LIBRARY 

OF  THE 

University  of  California. 

GIFT    OF 

Class 

INTRODUCTION 


ALGEBRA, 


BEING    THE 


FIRST    PART 


COURSE   OF    MATHEMATICS, 


TO    THE    METHOD    OF    INSTRUCTION 


AMERICAN    COLLEGES. 
Br   JEREMIAH    DAY,    D.D*LL.D*, 

LATE    PRESIDENT    OF    YALE    COLLEGE, 

A    NEW    EDITION.— Sixteenth  Thousand. 

WITH    ADDITIONS    AND    ALTERATIONS,    BY    THE   AUTHOR,    AXD    PROFESSOR    8TANLET 
Jty^*    IT*  YALE    COLiE^fe^k 

JEW    HAVEN  : 

DURRIE     <fc     PECK. 
PHILADELPHIA  \ 

H.   C.   PECK     k    T.   BLISS. 

1857. 


Do 


Entered  according  to  Act  of  Congress,  in  the  year  1852, 

By  Jeremiah  Day, 

in  the  Clerk's  Office  of  the  District  Court  of  Connecticut. 


^n. 


NEW    haven: 

PRINTKD      BY      EZgKIRL     HATB0. 


PREFACE. 


The  following  summary  view  of  the  first  principles  ^, 
algebra  is  intended  to  be  accommodated  to  the  method  o\ 
instruction  generally  adopted  in  the  American  colleges. 

The  books  which  have  been  published  in  Great  Britain  on 
mathematical  subjects,  are  principally  of  two  classes. — One 
consists  of  extended  treatises,  which  enter  into  a  thorough 
investigation  of  the  particular  departments  which  are  the 
objects  of  their  inquiry.  Many  of  these  are  excellent  in 
their  kind ;  but  they  are  too  voluminous  for  the  use  of  the 
body  of  students  in  a  college. 

The  other  class  are  expressly  intended  for  beginners ;  but 
many  of  them  are  written  in  so  concise  a  manner,  that  im- 
portant proofs  and  illustrations  are  excluded.  They  are  mere 
text-books,  containing  only  the  outlines  of  subjects  which  are 
to  be  explained  and  enlarged  upon,  by  the  professor  in  his 
lecture  room,  or  by  the  private  tutor  in  his  chamber. 

In  the  colleges  in  this  country,  there  is  generally  put  into 
the  hands  of  a  class,  a  book  from  which  they  are  expected  of 
themselves  to  acquire  the  principles  of  the  science  to  which 
they  are  attending :  receiving,  however,  from  their  instructor, 
any  additional  assistance  which  may  be  found  necessary.  An 
elementary  work  for  such  a  purpose,  ought  evidently  to  contain 
the  explanations  which  are  requisite,  to  bring  the  subjects 
treated  of  within  the  comprehension  of  the  body  of  the  class. 

If  the  design  of  studying  the  mathematics  ware  merely  to 
obtain  such  a  knowledge  of  the  practical  parts,  as  is  required 
for  transacting  business ;  it  might  be  sufficient  to  commit  to 
memory  some  of  the  principal  rules,  and  to  make  the  opera- 
tions familiar,  by  attending  to  the  examples.  In  this  mechan- 
ical way,  the  accountant,  the  navigator,  and  the  land  surveyor, 
may  be  qualified  for  their  respective  employments,  with  very 
little  knowledge  of  the  principles  that  lie  at  the  foundation 
of  the  calculations  which  they  are  to  make. 

But  a  higher  object  is  proposed,  in  the  case  of  those  who 
are  acquiring  a  Lberal  education.  The  main  design  should 
be  to  call  into  exercise,  to  discipline,  and  to  invigorate  the 
powers  of  the  mind.     It  is  the  logic  of  the  mathematics  which 


3v  PREFACK. 

constitutes  their  principal  value,  as  a  part  of  a  course  of  col- 
legiate instruction.  The  time  and  attention  devoted  to  them, 
is  for  the  purpore  of  forming  sound  reasoners,  rather  than 
expert  mathematicians.  To  accomplish  this  object  it  is  neces- 
sary that  the  principles  be  clearly  explained  and  demonstra- 
ted, and  that  the  several  parts  be  arranged  in  such  a  manner, 
as  to  show  the  dependence  of  one  upon  another.  The  whole 
should  be  so  conducted,  as  to  keep  the  reasoning  powers  in 
continual  exercise,  without  greatly  fatiguing  them.  No  other 
subject  affords  a  better  opportunity  for  exemplifying  the  rules 
of  correct  thinking.  A  more  finished  specimen  of  clear 
and  exact  logic  has,  perhaps,  never  been  produced,  than  the 
Elements  of  Geometry  by  Euclid. 

It  may  be  thought,  by  some,  to  be  unwise  to  form  our  gen- 
eral habits  of  arguing,  on  the  model  of  a  science  in  which 
the  inquiries  are  accompanied  with  absolute  certainty ;  while 
the  common  business  of  life  must  be  conducted  upon  probable 
evidence,  and  not  upon  principles  whrch  admit  of  complete 
demonstration.  There  would  be  weight  in  this  objection,  if 
the  attention  were  confined  to  the  pure  mathematics.  But 
when  these  are  connected  with  the  physical  sciences,  as- 
tronomy, chemistry,  and  natural  philosophy,  the  mind  has 
opportunity  to  exercise  its  judgment  upon  all  the  various 
degrees  of  probability  which  occur  in  the  concerns  of  life. 

So  far  as  it  is  desirable  to  form  a  taste  for  mathematical 
studies,  it  is  important  that  the  books  by  which  the  student  is 
first  introduced  to  an  acquaintance  with  these  subjects,  should 
not  be  rendered  obscure  and  forbidding  by  their  conciseness. 
Here  is  no  opportunity  to  awaken  interest,  by  rhetorical  ele- 
gance, by  exciting  the  passions,  or  by  presenting  images  to 
the  imagination.  The  beauty  of  the  mathematics  depends 
on  the  distinctness  of  the  objects  of  inquiry,  the  symmetry 
of  their  relations,  the  luminous  nature  of  the  arguments,  and 
the  certainty  of  the  conclusions.  But  how  is  this  beauty  to 
be  perceived,  in  a  work  which  is  so  much  abridged,  that  the 
chain  of  reasoning  is  often  interrupted,  important  demonstra- 
tions omitted,  and  the  transitions  from  one  subject  to  another 
so  abrupt,  as  to  keep  their  connections  and  dependencies  out 
of  view  ? 

It  may  not  be  necessary  to  state  every  proposition  and  its 
proof,  with  all  the  formality  which  is  so  strictly  adhered  to 
by  Euclid  ;  as  it  is  not  essential  to  a  logical  argument,  that 
it  be  expressed  in  regular  and  entire  syllogisms.     A  step  of 


PREFACE.  V 

a  demonstration  may  be  safely  omitted,  when  it  is  so  simple 
and  obvious,  that  no  one  possessing  a  moderate  acquaintance 
with  the  subject,  could  fail  to  supply  it  for  himself.  But  this 
liberty  of  omission  ought  not  to  be  extended  to  cases  in  which 
it  will  occasion  obscurity  and  embarrassment.  If  it  be  de- 
sirable to  give  opportunity  for  the  mind  to  display  and  enlarge 
its  powers,  by  surmounting  obstacles  ;  full  scope  may  be  found 
for  this  kind  of  exercise,  especially  in  the  higher  branches  of 
the  mathematics,  from  difficulties  which  will  unavoidably 
occur,  without  creating  new  ones  for  the  sake  of  perplexing. 

Algebra  requires  to  be  treated  in  a  more  plain  and  diffuse 
manner,  than  some  other  parts  of  the  mathematics  ;  because 
it  is  to  be  attended  to,  early  in  the  course,  while  the  mind  of 
the  learner  has  not  been  habituated  to  a  mode  of  thinking  so 
abstract,  as  that  which  will  now  become  necessary.  He  has 
also  a  new  language  to  learn,  at  the  same  time  he  is  settling  the 
principles  upon  wThich  his  future  inquiries  are  to  be  conducted. 
These  principles  ought  to  be  established,  in  the  most  clear  and 
satisfactory  manner  which  the  nature  of  the  case  will  admit  of. 
Algebra  and  geometry  may  be  considered  as  lying  at  the 
foundation  of  the  succeeding  branches  of  the  mathematics, 
both  pure  and  mixed.  Euclid  and  others  have  given  to  the  geo- 
metrical part  a  degree  of  clearness  and  precision  which  would 
be  very  desirable,  but* is  hardly  to  be  expected,  in  algebra. 

For  the  reasons  which  have  been  mentioned,  the  manner 
in  which  the  following  pages  are  written,  is  not  the  most 
concise.  But  the  work  is  necessarily  limited  in  extent  of 
subject.  It  is  far  from  being  a  complete  treatise  of  algebra. 
It  is  merely  an  introduction.  It  is  intended  to  contain  as  ' 
much  matter,  as  the  student  at  college  can  attend  to,  with 
advantage,  during  the  short  time  allotted  to  this  particular 
study.  There  is  generally  but  a  small  portion  of  a  class,  who 
have  either  leisure  or  inclination,  to  pursue  mathematical 
inquiries  much  farther  than  is  necessary  to  maintain  an 
honorable  standing  in  the  institution  of  which  they  are  mem- 
bers. Those  few  who  have  an  unusual  taste  for  this  science, 
and  aim  to  become  adepts  in  it.  ought  to  be  referred  to  sepa- 
rate and  complete  treatises,  on  the  different  branches.  No 
one  who  wishes  to  be  thoroughly  versed  in  mathematics, 
should  look  to  compendiums  and  elementary  books  for  any 
thing  more  than  the  first  principles.  As  soon  as  these  are 
acquired,  he  should  be  guided  in  his  inquiries  by  the  genius 
and  spirit  of  original  authors. 


VI  PREFACE. 

In  the  selection  of  materials,  those  articles  have  been  taken 
which  have  a  practical  application,  and  which  are  prepara- 
tory to  succeeiing  part-;  of  the  mathematics,  philosophy,  and 
astronomy.  The  object  has  not  been  to  introduce  original 
matter.  In  the  mathematics,  which  have  been  cultivated 
with  success  from  the  days  of  Pythagoras,  and  in  which  the 
principles  already  established  are  sufficient  to  occupy  the 
most  active  mind  for  years,  the  parts  to  which  the  student 
ought' first  to  attend,  are  not  those  recently  discovered.  Free 
use  has  been  made  of  the  works  of  Newton,  Maclaurin, 
Saunderson,  Simpson,  Euler,  Emerson,  Lacroix,  and  others, 
but  in  a  way  that  rendered  it  inconvenient  to  refer  to  them, 
in  particular  instances.  The  proper  field  for  the  display  of 
mathematical  genius,  is  in  tii3  region  of  invention.  But  what 
is  requisite  for  an  elementary  work,  is  to  collect,  arrange  and 
illustrate,  materials  already  provided.  However  humble  this 
employment,  he  ought  patiently  to  submit  to  it,  whose  object 
is  to  instruct,  not  those  who  have  made  considerable  progress 
in  the  mathematics,  but  those  who  are  just  commencing  the 
study.  Original  discoveries  are  not  for  the  benefit  of  begin- 
ners, though  they  may  be  of  great  importance  to  the  advance- 
ment of  science. 

The  arrangement  of  the  parts  is  such,  that  the  explanation 
of  one  is  not  made  to  depend  on  another  which  is  to  follow. 
In  the  statement  of  general  rules,  if  they  are  reduced  to  a 
small  number,  their  applications  to  particular  cases  may  not, 
always,  be  readily  understood.  On  the  other  hand,  if  they 
are  very  numerous,  they  become  tedious  and  burdensome  to 
the  memory.  The  rules  given  in  this  introduction,  are  most 
of  them  comprehensive;  but  they  are  explained  and  applied, 
in  subordinate  articles. 

A  particular  demonstration  is  sometimes  substituted  for  a 
general  one,  when  the  application  of  the  principle  to  other 
cases  is  obvious.  The  examples  are  not  often  taken  from 
philosophical  subjects,  as  the  learner  is  supposed  to  be  familiar 
with  none  of  the  sciences  except  arithmetic.  In  trea 
negative  quantities,  frequent  references  arc  made  to  m  \\ 
tile  concerns,  to  debt,  and  credit,  &c.  These  are  m  >T  Ay  for 
the  purpose  of  illustration.  The  whole  doctrine  of  negatives 
is  made  to  depend  on  th  }  single  principle,  that  they  are  quan- 
tities to  be  subtracted,  But  the  student,  at  this  early  period, 
is  not  accustomed  to  abstraction.  He  requires  particular 
examples,  to  catch  his  attention,  and  aid  his  conceptions. 


PREFACE.  Vll 


The  section  on  proportion,  will,  perhaps,  be  thought  use- 
less to  those  who  read  the  fifth  Book  of  Euclid.  That  is 
sufficient  for  the  purposes  of  pure  geometrical  demonstration. 
But  it  is  important  that  the  propositions  should  also  be  pre- 
sented under  the  algebraic  forms.  In  addition  to  this,  great 
assistance  may  be  derived  from  the  algebraic  notation,  in 
demonstrating,  and  reducing  to  system,  the  laws  of  propor- 
tion. The  subject  instead  of  being  broken  up  into  a  multi- 
tude of  distinct  propositions,  may  be  comprehended  in  a  few 
general  principles. 

THE    REVISED    EDITION. 

When  it  was  found  necessary  to  renew  the  stereotype 
plates  for  the  algebra,  which  were  too  much  impaired  to  be 
longer  used,  the  opportunity  was  embraced  to  make  additions 
and  alterations,  to  adapt  it  to  the  advance  which  had  been 
made,  in  this  department  of  collegiate  instruction,  since  the 
work  was  first  written.  Professor  Stanley  of  Yale  College 
was  applied  to,  to  make  the  proposed  revision.  He  had  pro- 
ceeded through  the  section  on  simple  equations,  when  it  was 
deemed  expedient,  that  he  should  suspend  his  professional 
engagements,  and  cross  the  Atlantic,  for  the  recovery  of  his 
health.  In  his  absence,  the  revision  was  continued  by  the 
author.  After  his  return,  Mr.  Stanley  made  the  important 
additions,  in  the  two  sections  on  the  general  properties  and 
solution  of  the  higher  equations.  It  is  not  practicable  to 
comprise,  in  a  single  volume  of  moderate  size,  the  entire  sci- 
ence and  art  of  algebra.  The  extent  of  a  text-book  on  the 
subject  must  be  proportioned  to  the  amount  of  time  which 
can  be  allotted  to  the  study  of  it,  without  encroaching  upon 
other  departments  of  instruction,  in  the  colleges,  and  other 
scientific  institutions. 

Some  of  the  add.tions  which  have  been  made,  in  the  pres- 
ent edition,  are  multiplication  and  division  by  detached  co- 
efficients, general  properties  of  quadratic  equations,  permu- 
tations and  combinations,  demonstrations  of  the  binomial 
theorem,  in  the  four  cases  of  integral,  fractional,  positive  and 
negative  exponents,  continued  fractions,  interpolation,  general 
properties  and  transformation  of  equations,  Sturm's  Theorem, 
and  Horner's  Method,  for  the  solution  of  the  higher  equa- 
tions. The  last  two  improvements  had  not  been  made  known, 
when  this  work  was  originally  published. 


CONTENTS. 


INTRODUCTORY     OBSERVATIONS. 

Pag* 

Primary  Departments  of  the  Mathematics, 1 

Definitions,  Axioms,  and  Demonstration, 2 

Practical  applications  of  the  Mathematics, 6 

Logic  of  the  Mathematics, 6 

SECTION    I. 
Notation,    Definitions,   Axioms,  <fec 

Definition  and  explanation  of  Algebra 9 

Notation  and  explanation  of  algebraic  Signs, 11 

Positive  and  Negative  quantities, 18 

Axioms,    21 

SECTION    II. 
Addition. 

General  rule  for  Addition,   24 

Reduction  of  like  terms, 25 

Reduction  when  the  signs  are  diiferent, 28 

SECTION    III. 

Subtraction. 

General  rule,   81 

SECTION    IV. 

Multiplication. 

Definition  and  explanation, 36 

Multiplication  of  Monomials 36 

Multiplication  of  Powers, 37 

Multiplication  of  Polynomials,    38 

Rule  for  the  Signs,    40 

Multiplication  by  detached  Co-efficients, 44 

Three  Theorems, -1 7 

SECTION    V. 
Division. 

Division  of  Monomials,    61 

Rule  for  the  Signs,    60 

Division  of  Polynomials, 52 

When  the  divisor  and  dividend  are  both  polynomials,    55 

Dividing  am-bm  by  a  - 1>,    60 

Division  by  Detached  Co-efficients, 61 

Value  of  the  Quotient,  68 

Resolving  polynonuals  into  i< actors,  65 


CONTENTS.,  IX 


SECTION   VI. 

Fractions.  Pa^ 

Value  of  a  Fraction, 67 

Application  of  the  signs  -f-  and   — , 69 

Reduction  of  Fractions 71 

Addition  and  Subtraction  of  Fractions, 73 

Multiplication  and  Division  of  Fractions,   78 

Fractional  numerators  and  denominators, 84 

SECTION    VII. 

Simple  Equations. 

Definitions  and  explanations, 86 

Reduction  of  equations,    87 

By  Transposition, 88 

By  Multiplication, 90 

By  Division, 93 

Clearing  equatious  from  fractions,    92 

Converting  proportions  into  equations,   96 

Solution  of  Problems, 97 

SECTION    VIII. 
Equations  containing  two  or  more  Unknown  Quantities. 

Elimination  by  Comparison, 108 

By  Substitution 109 

By  Addition  and  Subtraction, 110 

Three  or  more  unknown  quantities,   116 

General  rule  for  the  reduction  of  equations, 119 

Independent  Equations,    1 20 

Substituting  one  letter  for  several,   122 

Notation  by  primes,  seconds,  <fcc. 128 

Demonstration  of  Theorems,  125 

SECTION    IX. 
Involution  and  Evolution. 

Notation  of  Powers,    1 27 

Reciprocal  Powers, 129 

General  rule  for  Involution, 130 

-Rule  for  the  Signs  of  powers, 132 

Involving  a  Power,    132 

Powers  of  Fractions,   133 

Transferring  a  factor  in  a  fraction,   1 34 

Addition  and  Subtraction  of  powers,   135 

Multip'ication  and  Division  of  powers,    136 

Evolution,    139 

Notation  of  Roots, ; HO 

Power*  of  Roots, 142 

General  rule  for  Evolution, 145 

To  find  the  root  of  the  Product  of  factors,    14(* 

i.  Rule  for  the  Signs  of  the  roots 147 

^  Imaginary  quantities, 148 


X  CONTENTS. 

Page. 

Binomial  and  residual  square  roots,    151 

Surds,  or  radical  quantities,   152 

Reduction  of  radical  quantities, ; . .  152 

Addition  and  Subtraction  of  radical  quantities, 156 

Multiplication  and  Division  of  radical  quantities, 157 

Involution  and  Evolution  of  radical  quantities,    163 

Factors  producing  rational  products, 166 

Clearing  equations  from  radical  signs,    167 

SECTION    X. 

Quadratic   Equations. 

Reduction  of  equations  by  Involution  and  Evolution,    170 

Ambiguous  solutions,   1 74 

Affected  quadratic  equations,   177 

Completing  the  square, 179 

A  different  method  of  completing  the  square, 184 

Equations  of  the  form  x%n  +  xn= a, 186 

Positive  and  negative  results, 187 

<Roots  of  equations, 188 

^Imaginary  roots,   189 

Two  or  more  unknown  quantities,     190 

General  properties  of  quadratic  equations,    192 

Problems  producing  quadratic  equations, 196 

SECTION    XI. 

Ratio  and  Proportion. 

Arithmetical  Ratio, 201 

Geometrical  Ratio,   202 

Compound  Ratio,   204 

Multiplying  the  terms  of  a  ratio, 207- 

Adding  or  subtracting  terms, 209 

Ratios  of  Inequality,   211 

Proportion, 213 

Continued  Proportion,    215 

Geometrical  Proportion, 210 

Changing  the  order  of  the  terms, 218 

Multiplying  or  dividing  terms, 220 

Comparing  one  proportion  with  another,    221 

Addition  or  Subtraction  of  terms, 223 

Compounding  ratios, 2J5 

Powers  and  Roots  of  proportional  quantities, 286 

Continued  proportion, 

Harmonica!  proportion,    230 

SECTION    XII. 

Variation  or  General  Proportion. 

Direct  and  reciprocal  variation,    236 

Multiplying  and  dividing  terms 237 

Powers  and  roots  of  variable  quantities,    239 

Variable  and  constant  quantities, 239 


CONTENTS.  XI 

SECTION    XIII. 

Arithmetical  and  Geometrical  Progression. 

Page. 

Arithmetical  Progression, 241 

Formulas  for  the  first  and  last  terms,  <fec 243 

Sum  of  all  the  terms, 244 

Series  of  odd  numbers, 246 

Sums  or  differences,  products  or  quotients  of  arithmetical  series,   247 

Geometrical  Progression,   248 

Formulas  for  the  first  and  last  terms,  ratio,  <fcc 250 

Sum  of  all  the  terms,   251 

Harmonical  Progression, 252 

SECTION    XIV. 

Infinites  and  Infinitesimals. 

Mathematical  Infinity,    255 

Infinitesimals,    25(5 

Multiplication  and  Division  of  infinites,    259 

Value  of  the  expression  -» 260 

SECTION    XV. 

Common  Measure  and  Multiple,  Permutations  and  Combinations. 

Greatest  Common  Measure, 262 

Least  Common  Multiple,    265 

Permutations, * 266 

Combinations, 269 

SECTION    XVI. 

Involution  and  Expansion  of  Binomials. 

Powers  of  Binomials 272 

Co-efficients  of  the  terms,  273 

Law  of  the  co-efficients, 274 

Application  to  the  Powers  of  Binomials, 276 

Binomial  Theorem,   278 

Identical  equations, ^ 281 

Demonstration  for  a  fractional  exponent,   281 

For  a  negative  exponent 288 

Termination  of  series  from  binomials, 285 

Extracting  Roots  by  the  binomial  theorem, 286 

Factors  of  co-efficients  kept  distinct, 287 

SECTION    XVII. 

Evolution  of  Compound  Quantities. 

General  rule  for  extracting  roots,    289 

Rule  for  the  Square  Root,   291 

Roots  of  Binomial  Surds,   293 


Xll  CONTENTS. 

SECTION*    XVIII. 

Infinite  Series.  P 

Expanding  a  Fraction  into  a  series, 295 

Root  of  a  Compound  Surd, 297 

Indeterminate  Co-efficients,    298 

S  animation  of  Series, 300 

Recurring  Series, 301 

Scale  of  ^Relation, 302 

Method  of  Differences,   304 

Continued  Fractions, 308 

Converging  Fractions,  

Continued  Fractions  from  Surd  Quantities, 311 

Values  of  Continued  Fractions, 312 

Excess  or  Deficiency  of  a  Continued  Fraction, 314 

Interpolation, 315 

SECTION    XIX. 
General  Properties  of  Equations. 

Composition  of  Equations, 317 

Roots  of  Equations 318 

Equations  Divisible  by  Binomial  Factors, 319 

Number  of  the  Roots  of  an  Equation, 320 

Laws  of  the  Co-efficients  of  Equations,    322 

Positive  and  Negative  Roots 323  - 

Newton's  rule  for  the  co-efficients, 324 

Derived  Polynomials, 327 

Transformation  of  equations, 328 

Transformation  by  Successive  Divisions,    330 

Equations  wanting  the  Second  Term, 336 

Changing  the  Signs  of  alternate  terms 336 

The  first  term  made  greater  than  all  the  others, 338 

Imaginary  Roots  of  Equations,   341 

Variation  and  Permanence  of  signs,   '. 342 

Number  of  Positive  and  Negative  Roots, 343 

Equal  Roots  of  an  equation, 346 

JSturm's  Theorem,    849 

^Demonstration  of  Sturm's  Theorem,   ' 353 

Elimination  in  the  higher  equations, 357 

SECTION    XX. 
Resolution  of  Equations. 

Rational  Roots  of  equations, : 360 

Rule  for  finding  Integral  Roots, 362 

Solution  by  Double  Position, 365 

,'Newtons  Method,   368 

(Horner's  Method, 369 

SECTION    XXI. 

ArPLTCATION    OF    ALGEBRA    TO    GEOMETRY 

Positive  and  Negative  lines, 383 

Multiplication  of  Geometrical  Quantities, 383 

Algebraic  expressions  for  Areas,   885 

Contents  of  Solids 388 

Gei  mietrical  Problems,   3b9 

Notes    397 


INTRODUCTORY  OBSERVATIONS 


MATHEMATICS    IN    GENERAL. 



I 


Art.   1.  Mathematics  is  the  science  of  auANTiTY. 

Any  thing  which  can  be  multiplied,  divided,  or  measured,  is 
called  quantity.  Thus,  a  line  is  a  quantity,  because  it  can 
be  doubled,  trebled,  or  halved ;  and  can  be  measured,  by  ap- 
plying to  it  another  line,  as  a  foot,  a  yard,  or  an  ell.  Weight 
is  a  quantity,  which  can  be  measured,  in  pounds,  ounces,  and 
grains.  Time  is  a  species  of  quantity,  whose  measure  can  be 
expressed,  in  hours,  minutes,  and  seconds.  But  color  is  not 
a  quantity.  It  can  not  be  said,  with  propriety,  that  one  color 
is  twice  as  great,  or  half  as  great,  as  another.  The  opera- 
tions of  the  mind,  such  as  thought,  choice,  desire,  hatred,  &c. 
are  not  quantities.     They  are  incapable  of  mensuration.* 

Those  parts  of  the  Mathematics,  on  which  all  the  others 
are  founded,  are  Arithmetic,  Algebra,  and  Geometry. 

2,  Arithmetic  is  the  science  of  numbers.  Its  aid  is  re- 
quired to  complete  and  apply  the  calculations,  in  almost  every 
other  department  of  the  mathematics. 

3.  Algebra  is  a  method  of  computing  by  letters  and  other 
symbols.  Fluxions,  or  the  Differential  and  Integral  Calcu- 
lus, may  be  considered  as  belonging  to  the  higher  branches 
of  Algebra. 


*  See  Note  A. 
1 


1  INTRODUCTORY  OBSERVATIONS. 

4.  Geometry  is  that  part  of  the  mathematics,  which  treats 
of  magnitude.  By  magnitude,  in  the  appropriate  sense  of 
me  term,  is  meant  that  species  of  quantity,  which  is  extended; 
mat  is,  which  has  one  or  more  of  the  three  dimensions,  length, 
breadth,  and  thickness.  Thus  a  line  is  a  magnitude,  because 
it  is  extended,  in  length.  A  surface  is  a  magnitude,  having 
length  and  breadth.  A  solid  is  a  magnitude,  having  length, 
breadth,  and  thickness.  But  motion,  though  a  quantity,  is 
not,  strictly  speaking,  a  magnitude.  xIt  has  neither  length, 
breadth,  nor  thickness.* 

Trigonometry  and  Conic  Sections  are  branches  of  the 
mathematics,  in  which  the  principles  of  Geometry  are  applied 
to  triangles,  and  the  sections  of  a  cone. 

5.  Mathematics  are  either  pure  or  mixed.  In  pure  mathe- 
matics, quantities  are  considered,  independently  of  any  sub- 
stances actually  existing.  But,  in  mixed  mathematics,  the 
relations  of  quantities  are  investigated,  in  connection  with 
some  of  the  properties  of  matter,  or  with  reference  to  the 
common  transactions  of  business.  Thus,  in  Surveying, 
mathematical  principles  are  applied  to  the  measuring  of  land; 
in  Optics,  to  the  properties  of  light ;  and  in  Astronomy,  to 
the  motions  of  the  heavenly  bodies. 

The  science  of  the  pure  mathematics  has  long  been  distin- 
guished, for  the  clearness  and  distinctness  of  its  principles ; 
and  the  irresistible  conviction,  which  they  carry  to  the  mind 
of  every  one  who  is  once  made  acquainted  with  them.  This 
is  to  be  ascribed,  partly  to  the  nature  of  the  subjects,  and 
partly  to  the  exactness  of  the  definitions,  the  axioms,  and  the 
demonstrations. 

6.  The  foundation  of  all  mathematical  knowledge  must 
be  laid  in  definitions  and  self-evident  truths.  A  definition  is 
an  explanation  of  what  is  meant,  by  any  word  or  phrase. 
Thus,  an  equilateral  triangle  is  defined,  by  saying,  that  it  is  a 
figure  bounded  by  three  equal  sides. 

It  is  essential  to  a  complete  definition,  that  it  perfectly  dis- 
tinguish the  thing  defined,  from  every  thing  else.  On  many 
subjects  it  is  difficult  to  give  such  precision  to  language,  that 
it  shall  convey,  to  every  hearer  or  reader,  exactly  the  same 
ideas.  But  in  the  mathematics,  the  principal  terms  may  be 
so  defined,  as  not  to  leave  room  for  the  least  difference  of 


*  Some  writers,  however,  use  magnitude  as  synonymous  with  quantity. 


INTRODUCTORY   OBSERVATIONS.         3 

apprehension,  respecting  their  meaning.  All  must  be  agreed, 
as  to  the  nature  of  a  circle,  a  square,  and  a  triangle,  when 
they  have  once  learned  the  definitions  of  these  figures. 

Under  the  head  of  definitions,  may  be  included  explana- 
tions of  the  characters  which  are  used  to  denote  the  rela- 
tions of  quantities.  Thus  the  character  V  is  explained  or 
defined,  by  saying  that  it  signifies  the  same  as  the  words 
square  root. 

7.  The  next  step,  after  becoming  acquainted  with  thr 
meaning  of  mathematical  terms,  is  to  bring  them  together,  \v 
the  form  of  propositions.  Some  of  the  relations  of  quantities 
require  no  process  of  reasoning,  to  render  them  evident.  To 
be  understood,  they  need  only  to  be  proposed.  That  a  square 
is  a  different  figure  from  a  circle ;  that  the  whole  of  a  thing 
is  greater  than  one  of  its  parts ;  and  that  two  straight  lines 
can  not  enclose  a  space,  are  propositions  so  manifestly  true, 
that  no  reasoning  upon  them  could  make  them  more  certain. 
They  are,  therefore,  called  self-evident  truths,  or  axioms. 

8.  There  are,  however,  comparatively  few  mathematical 
truths  which  are  self-evident.  Most  require  to  be  proved  by 
a  chain  of  reasoning.  Propositions  of  this  nature  are  denom- 
inated theorems;  and  the  process,  by  which  they  are  shown 
to  be  true,  is  called  demonstration.  This  is  a  mode  of  argu- 
ing, in  which,  every  inference  is  immediately  derived,  either 
from  definitions,  from  axioms,  or  from  principles  which  have 
been  previously  demonstrated.  In  this  way,  complete  cer- 
tainty is  made  to  accompany  every  step,  in  a  long  course  of 
reasoning. 

9.  Demonstration  is  either  direct  or  indirect.  The  former 
is  the  common,  obvious  mode  of  conducting  a  demonstrative 
argument.  But  in  some  instances,  it  is  necessary  to  resort  to 
indirect  demonstration ;  which  is  a  method  of  establishing  a 
proposition,  by  proving  that  to  suppose  it  not  true,  would  lead 
to  an  absurdity.  This  is  frequently  called  reductio  ad  ab- 
surdum.  Thus,  in  certain  cases  in  geometry,  two  lines  may 
be  proved  to  be  equal,  by  showing  that  to  suppose  them  un- 
equal, would  involve  an  absurdity. 

10.  Besides  the  principal  theorems  in  the  mathematics, 
there  are  also  Lemmas  and  Corollaries. 

A  lemma  is  a  proposition  which  is  demonstrated,  for  the 
purpose  of  using  it,  in  the  demonstration  of  some  other  propo- 
sition.    This  preparatory  step  is  taken  to  prevent  the  proof 


4         INTRODUCTORY   OBSERVATIONS. 

of  the  principal  theorem  from  becoming  complicated  and 
tedious. 

A  corollary  is  an  inference  from  a  preceding  proposition. 
A  Scholium  is  a  remark  of  any  kind,  suggested  by  something 
which  has  gone  before,  though  not,  like  a  corollary,  immedi- 
ately depending  on  it. 

1 1 .  The  immediate  object  of  inquiry,  in  the  mathematics, 
is,  frequently,  not  the  demonstration  of  a  general  truth,  but  a 
method  of  performing  some  operation,  such  as  reducing  a 
vulgar  fraction  to  a  decimal,  extracting  the  cube  root,  or  in- 
scribing a  circle  in  a  square.  This  is  called  solving  a  prob- 
lem. A  theorem  is  something  to  be  proved.  A  problem  is 
something  to  be  done. 

When  that  which  is  required  to  be  done,  is  so  easy,  as  to 
be  obvious  to  every  one,  without  an  explanation,  it  is  called 
a  postulate.  Of  this  nature  is  the  drawing  of  a  straight  line, 
from  one  point  to  another. 

13.  A  quantity  is  said  to  be  given,  when  it  is  either  sup- 
posed to  be  already  known,  or  is  made  a  condition,  in  the 
statement  of  any  theorem  or  problem.  In  the  rule  of  pro- 
portion in  arithmetic,  for  instance,  three  terms  must  be  given 
to  enable  us  to  find  a  fourth.  These  three  terms  are  the  data, 
upon  which  the  calculation  is  founded.  If  we  are  required 
to  find  the  number  of  acres,  in  a  circular  island  ten  miles  in 
circumference,  the  circular  figure,  and  the  length  of  the  cir- 
cumference are  the  data.  They  are  said  to  be  given  by  sup- 
position, that  is,  by  the  conditions  of  the  problem.  A  quantity 
is  also  said  to  be  given,  when  it  may  be  directly  and  easily 
inferred  from  something  else  which  is  given.  Thus,  if  two 
numbers  are  given,  their  sum  is  given  ;  because  it  is  obtained, 
by  merely  adding  the  numbers  together. 

In  Geometry,  a  quantity  may  be  given,  either  in  position, 
or  magnitude,  or  both.  A  line  is  given  in  position,  when  its 
situation  and  direction  are  known.  It  is  given  in  magnitude, 
when  its  length  is  known.  A  circle  is  given  in  position, 
when  the  place  of  its  centre  is  known.  It  is  given  in  mag- 
nitude,  when  the  length  of  its  diameter  is  known. 

1  «t.  One  proposition  is  contrary  or  contradictory  to  anoth- 
er, when,  what  is  affirmed,  in  the  one,  is  denied,  in  the  other. 

A  proposition  and  its  contrary,  can  never  both  be  true.  It 
can  not  be  true,  that  two  given  lines  are  equal,  and  that  they 
are  not  equal,  at  the  same  time. 


INTRODUCTORY   OBSERVATIONS.         3 

14.  One  proposition  is  the  converse  of  another,  when  the 
order  is  inverted ;  so  that,  what  is  given  or  supposed  in  the 
first,  becomes  the  conclusion  in  the  last ;  and  what  is  given 
in  the  last,  is  the  conclusion,  in  the  first.  Thus,  it  can  be 
proved,  first,  that  if  the  sides  of  a  triangle  are  equal,  the  an- 
gles are  equal ;  and  secondly,  that  if  the  angles  are  equal,  the 
sides  are  equal.  Here,  in  the  first  proposition,  the  equality 
of  the  sides  is  given;  and  the  equality  of  the  angles  inferred: 
in  the  second,  the  equality  of  the  angles  is  given,  and  the 
equality  of  the  sides  inferred. 

In  many  instances,  a  proposition  and  its  converse  are  both 
true ;  as  in  the  preceding  example.  But  this  is  not  always 
the  case.  A  circle  is  a  figure  bounded  by  a  curve ;  but  a 
figure  bounded  by  a  curve  is  not  of  course  a  circle. 

1«>.  The  practical  applications  of  the  mathematics,  in  the 
common  concerns  of  business,  in  the  useful  arts,  and  in  the 
various  branches  of  physical  science  are  almost  innumerable. 
Mathematical  principles  are  necessary  in  Mercantile  transac- 
tions, for  keeping,  arranging,  and  settling  accounts,  adjusting 
the  prices  of  commodities,  and  calculating  the  profits  of  trade: 
in  Navigation,  for  directing  the  course  of  a  ship  on  the  ocean, 
adapting  the  position  of  her  sails  to  the  direction  of  the  wind, 
finding  her  latitude  and  longitude,  and  determining  the  bear- 
ings and  distances  of  objects  on  shore:  in  Surveying,  for 
measuring,  dividing,  and  laying  out  grounds,  taking  the  eleva- 
tion of  hills,  and  fixing  the  boundaries  of  fields,  estates  and 
public  territories :  in  Civil  Engineering,  for  constructing 
bridges,  aqueducts,  locks,  &c:  in  Mechanics,  for  understand- 
ing the  laws  of  motion,  the  composition  of  forces,  the  equili- 
brium of  the  mechanical  powers,  and  the  structure  of  ma- 
chines :  in  Architecture,  for  calculating  the  comparative 
strength  of  timbers,  the  pressure  which  each  will  be  required 
to  sustain,  the  forms  of  arches,  the  proportions  of  columns, 
&c. :  in  Fortification,  for  adjusting  the  position,  lines,  and  an- 
gles, of  the  several  parts  of  the  works  :  in  Gunnery,  for  regu- 
lating the  elevation  of  the  cannon,  the  force  of  the  powder, 
and  the  velocity  and  range  of  the  shot :  in  Optics,  for  tracing 
the  direction  of  the  rays  of  light,  understanding  the  forma- 
tion of  images,  the  laws  of  vision,  the  separation  of  colors,  the 
nature  of  the  rainbow,  and  the  construction  of  microscopes 
and  telescopes :  in  Astronomy,  for  computing  the  distances, 
magnitudes,  and  revolutions  of  the  heavenly  bodies ;  and  the 
influence  of  the  law  of  gravitation,  in  raising  the  tides,  dis- 

1* 


6         INTRODUCTORY  OBSERVATIONS. 

turbing  the  motions  of  the  moon,  causing  the  return  of  the 
comets,  and  retaining  the  planets  in  their  orbits :  in  Geogra- 
phy, for  determining  the  figure  and  dimensions  of  the  earth, 
the  extent  of  oceans,  islands,  continents,  and  countries ;  the 
latitude  and  longitude  of  places,  the  courses  of  rivers,  the 
height  of  mountains,  and  the  boundaries  of  kingdoms:  in 
History,  for  fixing  the  chronology  of  remarkable  events,  and 
estimating  the  strength  of  armies,  the  wealth  of  nations,  the 
value  of  their  revenues,  and  the  amount  of  their  population : 
and,  in  the  concerns  of  Government,  for  apportioning  taxes, 
arranging  schemes  of  finance,  and  regulating  national  ex- 
penses. The  mathematics  have  also  important  applications 
to  Chemistry,  Mineralogy,  Music,  Painting,  Sculpture,  and 
indeed  to  a  great  proportion  of  the  whole  circle  of  arts  and 
sciences. 

16.  It  is  true,  that,  in  many  of  the  branches  which  have 
been  mentioned,  the  ordinary  business  is  frequently  transact- 
ed, and  the  mechanical  operations  performed,  by  persons  who 
have  not  been  regularly  instructed  in  a  course  of  mathemat- 
ics. Machines  are  framed,  lands  are  surveyed,  and  ships  are 
steered,  by  men  who  have  never  thoroughly  investigated  the 
principles,  which  lie  at  the  foundation  of  their  respective  arts. 
The  reason  of  this  is  that  the  methods  of  proceeding,  in 
their  several  occupations,  have  been  pointed  out  to  them,  by 
the  genius  and  labor  of  others.  The  mechanic  often  works 
by  rules,  wThich  men  of  science,  have  provided  for  his  use, 
and  of  which  he  knows  nothing  more  than  the  practical  ap- 
plication. The  mariner  calculates  his  longitude  by  tables,  for 
which  he  is  indebted  to  mathematicians  and  astronomers  of 
no  ordinary  attainments.  In  this  manner,  even  the  abstruse 
parts  of  the  mathematics,  are  made  to  contribute  their  aid  to 
the  common  arts  of  life. 

17.  But  an  additional  and  more  important  advantage  to 
persons  of  liberal  education,  is  to  be  found,  in  the  enlarge- 
ment and  improvement  of  the  reasoning  powers.  The  mind, 
like  the  body,  acquires  strength  by  exertion.  The  art  of 
reasoning,  like  other  arts,  is  learned  by  practice.  It  is  per- 
fected, only  by  long  continued  exercise.  Mathematical  stu- 
dies are  peculiarly  fitted  for  this  discipline  of  the  mind.  They 
are  calculated  to  form  it  to  habits  of  fixed  attention ;  of  sa- 
gacity, in  detecting  sophistry;  of  caution,  in  the  admission 
of  proof;  of  dexterity  in  the  arrangement  of  arguments ;  and 


INTRODUCTORY   OBSERVATIONS.         7 

of  skill,  in  making  all  the  parts  of  a  long  continued  process 
tend  to  a  result,  in  which  the  truth  is  clearly  and  firmly 
established.  When  a  habit  of  close  and  accurate  thinking  is 
thus  acquired,  it  may  be  applied  to  any  subject,  on  which  a 
man  of  letters  or  of  business  may  be  called  to  employ  his 
talents.  "The  youth,"  says  Plato,  "who  are  furnished  with 
mathematical  knowledge,  are  prompt  and  quick,  at  all  other 
sciences." 

It  is  not  pretended,  that  an  attention  to  other  objects  of 
inquiry  is  rendered  unnecessary,  by  the  study  of  the  mathe- 
matics. It  is  not  their  office,  to  lay  before  us  historical  facts  ; 
to  teach  the  principles  of  morals ;  to  store  the  fancy  with 
brilliant  images ;  or  to  enable  us  to  speak  and  write  with 
rhetorical  vigor  and  elegance.  The  beneficial  effects  which 
they  produce  on  the  mind,  are  to  be  seen,  principally,  in  the 
regulation  and  increased  energy  of  the  reasoning  powers. 
These  they  are  calculated  to  call  into  frequent  and  vigorous 
exercise.  At  the  same  time,  mathematical  studies  may  be 
so  conducted,  as  not  often  to  require  excessive  exertion  and 
fatigue.  Beginning  with  the  more  simple  subjects,  and  as- 
cending gradually  to  those  which  are  more  complicated,  the 
mind  acquires  strength  as  it  advances ;  and  by  a  succession 
of  steps,  rising  regularly  one  above  another,  is  enabled  to 
surmount  the  obstacles  which  lie  in  its  way.  In  a  course  of 
mathematics,  the  parts  succeed  each  other  in  such  a  con- 
nected series,  that  the  preceding  propositions  are  preparatory 
to  those  which  follow.  The  student  who  has  made  himself 
master  of  the  former,  is  qualified  for  a  successful  investigation 
of  the  latter.  But  he  who  has  passed  over  any  of  the  ground 
superficially,  will  find  that  the  obstructions  to  his  future  pro- 
gress are  yet  to  be  removed.  In  mathematics  as  in  war,  it 
should  be  made  a  principle,  not  to  advance,  while  any  thing 
is  left  unconquered  behind.  It  is  important  that  the  student 
should  be  deeply  impressed  with  a  conviction  of  the  necessity 
of  this.  Neither  is  it  sufficient  that  he  understands  the  na- 
ture of  one  proposition  or  method  of  operation,  before  pro- 
ceeding to  another.  He  ought  also  to  make  himself  familiar 
with  every  step,  by  careful  attention  to  the  examples.  He 
must  not  expect  to  become  thoroughly  versed  in  the  science, 
by  merely  reading  the  main  principles,  rules,  and  observa- 
tions. It  is  practice  only,  which  can  put  these  completely  in 
his  possession.  The  method  of  studying  here  recommended, 
is  not  only  that  which  promises  success,  but  that  which  will 


8  INTRODUCTORY     OBSERVATIONS 

be  found,  in  the  end,  to  be  the  most  expeditious,  and  by  far 
the  most  pleasant.  While  a  superficial  attention  occ^ions 
perplexity  and  consequent  aversion ;  a  thorough  investiga- 
tion is  rewarded  with  a  high  degree  of  gratification.  The 
peculiar  entertainment  which  mathematical  studies  are  cal- 
culated to  furnish  to  the  mind,  is  reserved  for  those  who 
make  themselves  masters  of  the  subjects  to  which  their  at- 
tention is  called. 


Note.    The  principal  definitions,  theorems,  rules,  <fec.  which  it  is  necessary  to 
commit  to  memory,  are  distinguished  by  being  put  in  Italics  or  Capitals. 


ALGEBRA. 


SECTION   I. 

NOTATION,    DEFINITIONS,    NEGATIVE    QUANTITIES,    AXIOMS,    <feC< 

Art.  18.  Algebra  may  be  defined,  a  brief  and  gen- 
eral METHOD  OF  SOLVING  QUESTIONS  CONCERNING  NUMBERS*. 
BY    MEANS    OF    LETTERS,    AND    OTHER    SYMBOLS. 

This,  it  must  be  acknowledged,  is  an  imperfect  account  of 
the  subject ;  as  every  account  must  necessarily  be,  which  is 
comprised  in  the  compass  of  a  definition.  Its  real  nature  is 
to  be  learned,  rather  by  an  attentive  examination  of  its  parts, 
than  from  any  summary  description. 

19.  The  solutions  in  Algebra,  are  of  a  more  general  na- 
ture than  those  in  common  Arithmetic.  The  latter  relate  to 
particular  numbers  ;  the  former  to  whole  classes  of  quantities. 
On  this  account,  Algebra  has  been  termed  a  kind  of  universal 
Arithmetic.  The  generality  of  its  solutions  is  principally 
owing  to  the  use  of  letters,  instead  of  numeral  figures,  to  ex- 
press the  several  quantities  which  are  subjected  to  calculation. 
One  of  the  nine  digits,  invariably  expresses  the  same  number. 
The  figure  8  always  signifies  eight ;  the  figure  5,  five,  &c. 
And,  though  one  of  the  digits,  in  connection  with  others,  may 
have  a  local  value,  different  from  its  simple  value  when  alone ; 
yet  the  same  combination  always  expresses  the  same  number. 
Thus  263  has  one  uniform  signification.  And  this  is  the 
case  with  every  other  combination  of  figures.  In  Arithmetic, 
therefore,  when  a  problem  is  solved,  the  answer  is  limited  to 
the  particular  numbers  which  are  specified,  in  the  statement 
of  the  question.  But  an  Algebraic  solution  may  be  equally 
applicable  to  all  other  quantities  which  have  the  same  rela- 
tions.    For  in  Algebra,  a  letter  may  stand  for  any  quantity 


10  ALGEBRA. 

which  we  wish  it  to  represent.  Thus  b  may  be  put  for  2,  or 
10,  or  50,  or  1000.  It  must  not  be  understood  from  this, 
however,  that  the  letter  has  no  determinate  value.  Its  value 
is  fixed  for  the  occasion.  For  the  present  purpose,  it  remains 
unaltered.  But  on  a  different  occasion,  the  same  letter  may- 
be put  for  any  other  number. 

20.  A  calculation  may  also  be  greatly  abridged  by  the  use 
of  letters ;  especially  when  very  large  numbers  are  concerned. 
When  several  such  numbers  are  to  be  combined,  as  in  multi- 
plication, the  process  becomes  extremely  tedious.  But  a  sin- 
gle letter  may  be  put  for  a  large  number,  as  well  as  for  a  small 
one.  The  numbers  26347297,  68347823,  and  27462498,  for 
instance,  may  be  expressed  by  the  letters,  b,  c,  and  d.  The 
multiplying  them  together,  as  will  be  seen  hereafter,  will  be 
nothing  more  than  writing  them,  one  after  another,  in  the 
form  of  a  word,  and  the  product  will  be  simply  bed.  This 
indeed  is  indicating  rather  than  performing  the  multiplication. 
But  it  is  often  sufficient  thus  to  indicate  what  is  to  be  done, 
without  executing  the  work.  It  may  happen,  in  the  solution 
of  a  problem,  that  a  multiplication  at  one  time  will  be  coun- 
terbalanced by  a  subsequent  division,  and  the  trouble  of  per- 
forming the  two  operations  will  be  saved,  if  the  first  be  only 
indicated,  till  its  effect  is  destroyed  by  the  other.  Solutions 
in  Algebra  are  sometimes  effected,  in  the  compass  of  a  few 
lines,  which,  in  common  Arithmetic,  must  be  extended  through 
many  pages. 

21.  Another  advantage  obtained  from  the  notation  by 
letters  instead  of  figures,  is,  that  the  several  quantities  which 
are  brought  into  calculation,  may  be  preserved  distinct  from 
each  other,  though  carried  through  a  number  of  complicated 
processes ;  whereas,  in  Arithmetic,  they  are  so  blended  to- 
gether, that  no  trace  is  left  of  what  they  were,  before  the 
operation  began. 

22.  Algebra  differs  farther  from  Arithmetic,  in  making 
use  of  unknown  quantities,  in  carrying  on  its  operations.  In 
Arithmetic,  all  the  quantities  which  enter  into  a  calculation 
must  be  known.  For  they  are  expressed  in  numbers.  And 
every  number  must  necessarily  be  a  determinate  quantity. 
But  in  Algebra,  a  letter  may  be  put  for  a  quantity,  before  its 
value  has  been  ascertained.  And  yet  it  may  have  such  rela- 
tions to  other  quantities,  with  which  it  is  connected,  as  to 
answer  an  important  purpose  in  the  calculation. 


NOTATION      AND      DEFINITIONS.  11 


NOTATION    AND   DEFINITIONS. 

33.  To  facilitate  the  investigations  in  algebra,  the  several 
steps  of  the  reasoning,  instead  of  being  expressed  in  words, 
are  translated  into  the  language  of  signs  and  symbols,  which 
may  be  considered  as  a  species  of  short-hand.  This  serves 
to  place  the  quantities  and  their  relations  distinctly  before  the 
eye,  and  to  bring  them  all  into  view  at  once.  They  are  thus 
more  readily  compared  and  understood,  than  when  removed 
at  a  distance  from  each  other,  as  in  the  common  mode  of 
writing.  But  before  any  one  can  avail  himself  of  this  advan- 
tage, he  must  become  perfectly  familiar  with  the  new  language. 

34.  The  quantities  in  algebra,  as  has  been  already  ob- 
served, are  generally  expressed  by  letters.  The  first  letters 
of  the  Alphabet  are  used  to  represent  known  quantities ;  and 
the  last  letters,  those  which  are  unknown.  Sometimes  the 
known  quantities,  instead  of  being  expressed  by  letters,  are 
set  down  in  figures,  as  in  common  arithmetic. 

3«i.  Besides  the  letters  and  figures,  there  are  certain  char- 
acters used,  to  indicate  the  relations  of  the  quantities,  or  the 
operations  which  are  performed  with  them. 

The  signs  +  and  — ,  which  are  read  plus  and  minus,  or 
more  and  less,  are  employed  to  denote  addition  and  subtrac- 
tion. Thus  a+b  signifies  that  b  is  to  be  added  to  a.  It  is 
read  a  plus  b,  or  a  added  to  b,  or  a  and  b.  If  the  expression 
be  a—b,  that  is,  a  minus  b;  it  indicates  that  b  is  to  be  sub- 
tracted from  a. 

36.  Quantities  having  the  sign  +  prefixed  are  called  posi- 
tive, and  those  which  have  the  sign  — ,  negative  quantities. 
For  the  nature  of  this  distinction,  see  Art.  49. 

All  the  quantities  which  enter  into  an  algebraic  process, 
are  considered,  for  the  purposes  of  calculation,  as  either  posi- 
tive or  negative.  Before  the  first  one,  unless  it  be  negative, 
the  sign  is  generally  omitted.  But  it  is  always  to  be  under- 
stood.    Thus  a-\-b,  is  the  same  as  -\-a-\-b. 

37.  Sometimes  both  +  and  —  are  prefixed  to  the  same 
letter.  The  sign  is  then  said  to  be  ambiguous.  Thus  a  ±  b 
signifies  that  in  certain  cases,  comprehended  in  a  general  so- 
lution, b  is  to  be  added  to  a,  and  in  other  cases  subtracted 
from  it. 


12  NOTATION      AND      DEFINITIONS. 

28.  When  it  is  intended  to  express  the  difference  between 
two  quantities,  without  deciding  which  is  the  one  to  be  sub- 
tracted, the  character  >  is  used.  Thus  a  ^b,  denotes  the 
difference  between  a  and  b,  without  determining  whether  a 
is  to  be  subtracted  from  b,  or  b  from  a. 

29.  The  character  X  denotes  multiplication.  Thus  aXb 
is  a  multiplied  into  b;  and  6X3  is  6  times  3,  or  6  into  3. 
Sometimes  a  point  is  used  to  indicate  multiplication.  Thus 
a.b  is  the  same  as  a X b.  But  the  multiplication  of  quantities 
that  are  represented  by  letters,  is  more  commonly  indicated 
by  connecting  the  letters  together  in  the  form  of  a  word  or 
syllable.  Thus  ab  is  the  same  as  a.b  or  aXb.  And  bcde  is 
the  same  as  bXcXdXe. 

When,  however,  the  quantities  are  expressed  numerically, 
the  sign  of  multiplication  must  not  be  omitted.  If  35  were 
written  for  the  product  of  3  into  5,  it  might  be  mistaken  for 
thirty-five. 

30.  When  two  or  more  quantities  are  multiplied  together, 
each  of  them  is  called  a  factor.  In  the  product  ab,  a  is  a 
factor,  and  so  is  b. 

A  quantity  is  said  to  be  .resolved  into  factors,  when  any 
factors  are  taken,  which,  being  multiplied  together,  will  pro- 
duce the  given  quantity.  Thus  ab  may  be  resolved  into  the 
two  factors  a  and  b,  because  aXb  is  ab.  And  amn  may  be 
resolved  into  the  three  factors  a,  and  m,  and  n.  And  48  may 
be  resolved  into  the  two  factors  2X24,  or  3X  16,  or  4  X  12,  or 
6X8;  or  into  the  three  factors  2X3X8,  or  4X6X2,  &c. 

3 1  •  A  numeral  figure  is  often  prefixed  to  a  letter.  This 
is  called  a  co-efficient.  It  shows  how  often  the  quantity  ex- 
pressed by  the  letter  is  to  be  taken.  Thus  2b  signifies  twice 
b ;  and  9b,  9  times  b,  or  9  multiplied  into  b. 

The  co-efficient  may  be  either  a  whole  number  or  a  frac- 
tion. Thus  §6  is  two-thirds  of  b.  When  the  co-efficient  is 
not  expressed,  1  is  always  to  be  understood.  Thus  a  is  the 
same  as  la ;  that  is,  once  a. 

The  co-efficient  may  be  a  letter,  as  well  as  a  figure.  In 
the  quantity  mb,  m  may  be  considered  the  co-efficient  of  b ; 
because  b  is  to  be  taken  as  many  times  as  there  are  units  in 
m.  If  m  stands  for  6,  then  mb  is  6  times  b.  In  Sabc,  3  may 
be  considered  as  the  co-efficient  of  abc ;  3a  the  co-efficient 
of  be ;  or  Sab,  the  co-efficient  of  c.  Every  co-efficient  is  a 
factor.     (Art.  30.) 


NOTATION      AND      DEFINITIONS.  13 

32.  The  character  ~  is  used  to  show  that  the  quantity 
which  precedes  it,  is  to  be  divided,  by  that  which  follows. 
Thus  fl-rc  is  a  divided  by  c;  and  ab-^-cd  is  the  product  of  a 
and  b,  divided  by  the  product  of  c  and  d. 

But  in  algebra,  division  is  more  commonly  expressed,  by 
writing  the  divisor  under  the  dividend,  in  the  form  of  a  vulgar 

fraction.     Thus  ■?  is  the  same  as  a+b;  and  -7—7  isthedifFer- 
b  d+h 

ence  of  c  and  b,  divided  by  the  sum  of  d  and  h. 

A  character  prefixed  to  the  dividing  line  of  a  fractional 

expression,  is  to  be  understood  as  referring  to  all  the  parts 

taken  collectively;  that  is  to  the  whole  value  of  the  quotient. 

b+c 
Thus  a ; —  signifies  that  the  quotient  of  b+c  divided  by 

,         ,  ,  c  ,  c—d      h+n   . 

m+n,  is  to  be  subtracted  irom  a.     And  — ■ —  X denotes 

a-\-m      x—y 

that  the  first  quotient  is  to  be  multiplied  into  the  second. 

33*  The  equality  between  two  quantities  or  sets  of  quan- 
tities is  expressed  by  parallel  lines  = .  Thus  a+b=d  signifies 
that  a  and  b  together  are  equal  to  d.  And  a+d=c=b-\-g=h, 
signifies  that  a  and  d  equal  c,  which  is  equal  to  b  and  g,  which 
are  equal  to  h.     So    8+4=16-4=10+2=7+2+3=12. 

34.  The  inequality  of  two  quantities  is  indicated  by  pla- 
cing the  character  >  between  them.  Thus  a>b  signifies  that 
a  is  greater  than  b. 

If  the  first  quantity  is  less  than  the  other,  the  character  < 
is  used ;  as  #<&;  that  is,  a  is  less  than  b.  In  both  cases,  the 
quantity  towards  which  the  character  opens,  is  greater  than 
the  other. 

3«>.  When  four  quantities  are  proportional,  the  proportion 
is  expressed  by  points,  in  the  same  manner,  as  in  the  Rule  of 
Three  in  arithmetic.  Thus  a\b\\c\  d  signifies  that  a  has  to 
b,  the  same  ratio  which  c  has  to  d.  And  ab  \  cd\\  a+m  ]  b+n, 
means,  that  ab  is  to  cd;  as  the  sum  of  a  and  m,  to  the  sum 
of  b  and  n. 

Three  points  .*.  are  sometimes  used  to  signify  therefore  or 
consequently. 

36.  A  power  of  a  quantity  is  the  product  formed  by  mul- 
tiplying the  quantity  into  itself.  Thus  2X2  or  4  is  the  square 
or  the  second  power  of  2;  2X2X2  or  8  is  the  cube  or  the 
third  power  of  2;  and  2X2X2X2  or  16,  the  fourth  power. 

2 


14  NOTATION      AND      DEFINITIONS. 

So  aXa  or  aa  is  the  second  power  of  a ;  aXaX a  or  aaa,  the 
third  power ;  aaaa,  the  fourth  power ;  &c. 

The  original  quantity  itself,  though  not,  like  the  powers  pro- 
ceeding from  it,  produced  by  multiplication,  is  nevertheless 
called  the  first  power.  It  is  also  called  the  root  of  the  other 
powers,  because  it  is  that  from  which  they  are  all  derived. 

37.  As  it  is  inconvenient,  especially  in  the  case  of  high 
powers,  to  write  down  all  the  letters  or  factors  of  which  the 
powers  are  composed,  an  abridged  method  of  notation  is  gen- 
erally adopted.  The  root  is  written  only  once ;  and  then  a 
number  or  letter  is  placed  at  the  right  hand,  and  a  little  ele- 
vated, to  signify  how  many  times  the  root  is  employed  as  a 
factor,  to  produce  the  power.  This  number  or  letter  is  called 
the  index  or  exponent  of  the  power.  Thus  a2  is  put  for  a X  a 
or  aa,  because  the  root  a,  is  twice  repeated  as  a  factor,  to 
produce  the  power  aa.  And  a3  stands  for  aaa;  for  here  a 
is  repeated  three  times  as  a  factor. 

The  index  of  the  first  power  is  1 ;  but  this  is  commonly 
omitted.     Thus  a1  is  the  same  as  a. 

Exponents  must  not  be  confounded  with  co-efficients.  A 
co-efficient  shows  how  often  a  quantity  is  taken  as  a  part  of 
a  whole.  An  exponent  shows  how  often  a  quantity  is  taken 
as  a  factor  in  a  product. 

Thus  4a=a+a+a+a.         But  a4=aXaXaXa. 

38.  A  simple  quantity  is  either  a  single  letter  or  number, 
or  several  letters  connected  together  without  the  signs  + 
and  —  .  Thus  a,  ab,  abd  and  Sb  are  each  of  them  simple 
quantities.  A  simple  quantity  is  also  called  a  monomial  or  a 
term. 

A  compound  quantity  consists  of  a  number  of  terms  or 
simple  quantities  connected  by  the  sign  +  or  — •  .  Thus  a +b, 
d—y,  b  —  d-\-3h,  are  each  compound  quantities.  Compound 
quantities  are  often  called  polynomials. 

If  there  are  two  terms  in  a  compound  quantity,  it  is  com- 
monly called  a  binomial.  Thus  a+ b  and  a— b  are  binomials. 
The  latter  is  also  called  a  residual  quantity,  because  it  ex- 
presses the  difference  of  two  quantities,  or  the  remainder, 
after  one  is  taken  from  the  other.  A  compound  quantity 
consisting  of  three  terms,  is  sometimes  called  a  trinomial; 
one  of  four  terms,  a  quadrinomial,  &c. 

39.  Similar  or  like  terms  are  those  in  which  the  letters 
are  the  same,  and  have  the  same  exponents.     And  unlike 


NOTATION      AND      DEFINITIONS.  15 

terms  are  those  in  which  the  letters  are  different,  or  the  same 
letter  has  different  exponents.  Thus  a2b,  3a2b,  -—a2b,  and 
—  6a2  b  are  like  terms,  because  their  letters  and  exponents  are 
the  same,  although  the  signs  and  co-efficients  are  different. 
But  the  terms  Sab,  3a2 b,  and  3axy  are  unlike,  because  they 
have  not  the  same  letters  and  exponents,  although  there  is  no 
difference  in  the  signs  and  co-efficients. 

40.  Every  letter  that  occurs  as  a  factor  in  a  term,  is 
called  a  dimension  of  that  term.  And  the  degree  of  a  term 
answers  to  the  number  of  its  dimensions.  A  numeral  co- 
efficient is  not  reckoned  as  a  dimension.  Thus,  2a  is  a  term 
of  one  dimension,  or  of  the  first  degree;  —3ab  is  a  term  of 
two  dimensions,  or  of  the  second  degree;  and  5ab2c2,  which 
is  the  same  as  babbbcc,  is  a  term  of  six  dimensions  or  of  the 
sixth  degree. 

The  degree  of  a  term,  or  the  number  of  its  dimensions,  is 
marked  by  the  sum  of  the  exponents  of  the  letters  contained 
in  the  term.  Thus,  the  number  of  dimensions  of  the  term 
6ab3cd2  is  1+3  +  1+2,  or  7. 

A  polynomial  is  said  to  be  homogeneous,  when  all  its  terms 
are  of  the  same  degree.  Thus,  labc—2x2-\-3xy2  is  homo- 
geneous: but  3ab+2ab2  — -4cx  is  not  homogeneous. 

41.  When  the  several  members  of  a  compound  quantity 
are  to  be  subjected  to  the  same  operation,  they  are  frequently 
connected  by  a  line  called  a  vinculum.  Thus  a—b  +  c  shows 
that  the  sum  of  b  and  c  is  to  be  subtracted  from  a.  But 
a—b+c  signifies  that  b  only  is  to  be  subtracted  from  a,  while 
c  is  to  be  added.  The  sum  of  c  and  d,  subtracted  from  the 
sum  of  a  and  b,  is  a+b— c+d.  The  marks  used  for  paren- 
theses, (),  are  often  substituted  instead  of  a  line  for  a  vincu- 
lum.    Thus  x—  (<z+c)  is  the  same  as  x—a+c. 

The  equality  of  two  sets  of  quantities  is  expressed,  without 
using  a  vinculum.  Thus  a-\-b=c+d  signifies,  not  that  b  is 
equal  to  c;  but  that  the  sum  of  a  and  b  is  equal  to  the  sum 
of  c  and  d. 

4&.  One  quantity  is  said  to  be  a  multiple  of  another,  when 
the  former  contains  the  latter  a  certain  number  of  times  with- 
out a  remainder.  Thus  \0a  is  a  multiple  of  2a  ;  and  24  is  a 
multiple  of  6. 

One  quantity  is  said  to  be  a  measure  of  another,  when  the 
former  is  contained  in  the  latter,  any  number  of  times,  with- 


16  NOTATION      AND      DEFINITIONS. 

out  a  remainder.     Thus  Sb  is  a  measure  of  156;  and  7  is  a 
measure  of  35. 

43.  The  reciprocal  of  a  quantity,  is  the  quotient  arising 
from  dividing  a  unit  by  that  quantity.    Thus  the  reciprocal  of 

a  is  - ;  the  reciprocal  of  a+b  is  —rr',  the  reciprocal  of  4  is  -. 

44.  A  single  letter,  or  a  number  of  letters,  representing 
any  quantities  with  their  relations,  is  called  an  algebraic  ex- 
pression; and  sometimes  a  formula.  Thus  a +b+ 3d  is  an 
algebraic  expression. 

The  value  of  an  expression,  is  the  number  or  quantity  for 
which  the  expression  stands.  Thus  the  value  of  3+4  is  7 ; 
of  3X4  is  12;  of  ^  is  2. 

45.  The  relations  of  quantities,  which  in  ordinary  lan- 
guage are  signified  by  words,  are  represented  in  the  alge- 
braic notation,  by  signs.  The  latter  mode  of  expressing  these 
relations  ought  to  be  made  so  familiar  to  the  mathematical 
student,  that  he  can,  at  any  time,  substitute  the  one  for  the 
other. 

A  few  examples  are  here  added,  in  which,  words  are  to  be 
converted  into  signs. 

1.  What  is  the  algebraic  expression  for  the  following  state- 
ment, in  which  the  letters  a,  b,  c,  &c.  may  be  supposed  to 
represent  any  given  quantities  ? 

The  product  of  a,  b  and  c,  divided  by  the  difference  of  c 
and  d,  is  equal  to  the  sum  of  b  and  c  added  to  15  times  A. 

abc      .  . 

Ans.     =  o+c+15A. 

2.  The  quotient  of  a  divided  by  ten  times  the  square  of  6, 
diminished  by  twice  the  product  of  c  and  d,  is  equal  to  the 
quotient  of  6  divided  by  the  difference  of  a  and  d.     Ans. 

3.  The  sum  of  a,  b  and  c,  divided  by  the  fifth  power  of  d, 
is  equal  to  4  times  the  difference  of  b  and  c,  added  to  9  times 
their  product.     Ans. 

4.  The  product  of  a,  b  and  c,  is  to  the  quotient  of  a  divided 
by  the  difference  of  b  and  c,  as  half  the  sum  of  a  and  A,  is  to 
4  times  the  difference  of  c  and  d.     Ans. 

5.  The  difference  between  twice  the  cube  of  a  and  the  pro- 
duct of  b  into  c,  is  equal  to  the  sum  of  a  and  c,  added  to  the 
quotient  of  b  divided  by  the  sum  of  d,  A  and  m.     Ans. 


NOTATION.  17 

46.  It  is  necessary  also,  to  be  able  to  reverse  what  is  done 
in  the  preceding  examples,  that  is,  to  translate  the  algebraic 
signs  into  common  language. 

What  will  the  following  expressions  become,  when  words 
are  substituted  for  the  signs. 

1.     ~t—=(20c— 6m- 


h  a-\-c' 

Arts.  The  sum  of  a  and  b  divided  by  A,  is  equal  to  the 
product  of  <z,  b  and  c,  diminished  by  6  times  m,  and  increased 
by  the  quotient  of  a  divided  by  the  sum  of  a  and  c. 

rt      «/    .      ,x  .     n       a*  —  bd 

Sabc        x  „ 

4.     =  -—  -  a(b-5)+2ch. 

(a+d)(b^c)      4c  A4  b+c+d 


a+mn         a+b    Sa(b+c)     a—(x+y)' 

47.  At  the  close  of  an  algebraic  process,  it  is  frequently 
necessary  to  restore  the  numbers,  for  which  letters  had  been 
substituted,  at  the  beginning.  In  doing  this,  the  sign  of  multi- 
plication must  not  be  omitted,  as  it  generally  is,  between  fac- 
tors, expressed  by  letters.  Thus,  if  a  stands  for  3,  and  b  for  4 ; 
the  product  ab  is  not  34,  but  3X4,  that  is,.  12.     See  Art.  29. 

In  the  following  examples, 

Let  a=S  And  d=6 

b=4  m=8 

c=2  7i=lG. 


Then,      1. 


a±m    be— n  _3+8    4X2—10 
cd  3d    ""2X6"1      3X6     " 

m-(b+e)     3(c+n) 

6n2    t    w       ..  7.     tan-bm 


-+5b(c+d-a)  + 


m  —  a         v  '      b3—2ac 

48.  An    algebraic  expression,   in  which  numbers   have 
been  substituted  for  letters,  may  often  be  rendered  much 

2* 


18       POSITIVE      AND      NEGATIVE      ftUANTITIES. 

more  simple,  by  reducing  several  terms  to  one.  This  can 
not  generally  be  done,  while  the  letters  remain.  If  a+b  is 
used  for  the  sum  of  two  quantities,  a  can  not  be  united  in 
the  same  term  with  b.  But  if  a  stands  for  3,  and  b  for  4,  then 
<z+&=3+4=7.  The  value  of  an  expression,  consisting  of 
many  terms  may  thus  be  found,  by  actually  performing,  with 
the  numbers,  the  operations  of  addition,  subtraction,  multipli- 
cation, &c.  indicated  by  the  algebraic  characters. 

Find  the  value  of  the  following  expressions,  in  which  the 
letters  are  supposed  to  stand  for  the  same  numbers,  as  in  the 
preceding  Article. 

ad  3X6 

1.  — +a+77i7i=-— +3+8X  10=9+3+80=92. 

c  2 

m-hn       /7       ,       2ac      8+10       ,  x  ,  2X3X2 

2.  r +6(ab-d)  + =  ^_+6(3X4-6)  +  irs~5-  = 

b— a       v  n—m     4—3        v  10—8 

v     5d2  —  (d—c)         mn 

3.  2(a+5_c)  +  __^____= 

■       x  .        *     mid— a)     6b  +  3n 

4.  (a+6+c)(n_£0+_L_^+__B3 

5b(2m+5a)         7(ab-4c)  _ 
5#     3(rf-c)  (3a-77i)  +  4(ll-2^)~ 


Positive  and  Negative  Quantities. 

49.  To  one  who  has  just  entered  on  the  study  of  algebra, 
there  is  generally  nothing  more  perplexing,  than  the  use  of 
what  are  called  negative  quantities.  He  supposes  he  is  about 
to  be  introduced  to  a  class  of  quantities  which  are  entirely 
new ;  a  sort  of  mathematical  nothings,  of  which  he  can  form 
no  distinct  conception.  As  positive  quantities  are  real,  he 
concludes  that  those  which  are  negative  must  be  imaginary. 
But  this  is  owing  to  a  misapprehension  of  the  term  negative, 
as  used  in  the  mathematics. 

50.  A  negative  quantity  is  one  which  is  required  to  be 
subtracted.  When  several  quantities  enter  into  a  calcula- 
tion, it  is  frequently  necessary  that  some  of  them  should  be 
added  together,  while  others  are  subtracted.  The  former  are 
called  affirmative  or  positive,  and  are  marked  with  the  sign  +  ; 


POSITIVE     AND      NEGATIVE      QUANTITIES.       19 

the  latter  are  termed  negative,  and  distinguished  by  the 
sign  —  .  If,  for  instance,  the  profits  of  trade  are  the  subject 
of  calculation,  and  the  gain  is  considered  positive;  the  loss 
will  be  negative,  because  the  latter  must  be  subtracted  from 
the  former,  to  determine  the  clear  profit.  If  the  sums  of  a 
book  account  are  brought  into  an  algebraic  process,  the  debt 
and  the  credit  are  distinguished  by  opposite  signs.  If  a  man 
on  a  journey  is,  by  any  accident,  necessitated  to  return  several 
miles,  this  backward  motion  is  to  be  considered  negative,  be- 
cause that,  in  determining  his  real  progress,  it  must  be  sub- 
tracted from  the  distance  which  he  has  travelled  in  the  oppo- 
site direction.  If  the  ascent  of  a  body  from  the  earth  be 
called  positive,  its  descent  will  be  negative.  These  are  only 
different  examples  of  the  same  general  principle.  In  each  of 
the  instances,  one  of  the  quantities  is  to  be  subtracted  from 
the  other. 

51.  The  terms  positive  and  negative,  as  used  in  the 
mathematics,  are  merely  relative.  They  imply  that  there  is, 
either  in  the  nature  of  the  quantities,  or  in  their  circumstan- 
ces, or  in  the  purposes  which  they  are  to  answer  in  calcula- 
tion, some  such  opposition  as  requires  that  one  should  be 
subtracted  from  the  other.  But  this  opposition  is  not  that  of 
existence  and  non-existence,  nor  of  one  thing  greater  than 
nothing,  and  another  less  than  nothing.  For,  in  many  cases, 
either  of  the  signs  may  be,  indifferently  and  at  pleasure,  ap- 
plied to  the  very  same  quantity;  that  is,  the  two  characters 
may  change  places.  In  determining  the  progress  of  a  ship, 
for  instance,  her  easting  may  be  marked  +  ,  and  her  west- 
ing —  ;  or  the  westing  may  be  +,  and  the  easting  — .  All  that 
is  necessary  is,  that  the  two  signs  be  prefixed  to  the  quanti- 
ties, in  such  a  manner  as  to  show,  which  are  to  be  added, 
and  which  subtracted.  In  different  processes,  they  may  be 
differently  applied.  On  one  occasion,  a  downward  motion 
may  be  called  positive,  and  on  another  occasion  negative. 

52.  In  every  algebraic  calculation,  some  one  of  the  quan- 
tities must  be  fixed  upon,  to  be  considered  positive.  All  other 
quantities  which  will  increase  this,  must  be  positive  also. 
But  those  which  will  tend  to  diminish  it,  must  be  negative. 
In  a  mercantile  concern,  if  the  stock  is  supposed  to  be  posi- 
tive, the  profits  will  be  positive  ;  for  they  increase  the  stock ; 
they  are  to  be  added  to  it.  But  the  losses  will  be  negative ; 
for  they  diminish  the  stock ;  they  are  to  be  subtracted  from  it. 


20  NEGATIVE      QUANTITIES. 

When  a  boat,  in  attempting  to  ascend  a  river,  is  occasionally 
driven  back  by  the  current ;  if  the  progress  up  the  stream,  to 
any  particular  point,  is  considered  positive,  every  succeeding 
instance  of  forward  motion  will  be  positive,  while  the  back- 
ward motion  will  be  negative. 

53.  A  negative  quantity  is  frequently  greater,  than  the 
positive  one  with  which  it  is  connected.  But  how,  it  may 
be  asked,  can  the  former  be  subtracted  from  the  latter  ?  The 
greater  is  certainly  not  contained  in  the  less :  how  then  can 
it  be  taken  out  of  it  ?  The  answer  to  this  is,  that  the  greater 
may  be  supposed  first  to  exhaust  the  less,  and  then  to  leave  a 
remainder  equal  to  the  difference  between  the  two.  If  a 
man  has  in  his  possession  1000  dollars,  and  has  contracted  a 
debt  of  1500 ;  the  latter  subtracted  from  the  former,  not  only 
exhausts  the  whole  of  it,  but  leaves  a  balance  of  500  against 
him.  In  common  language,  he  is  500  dollars  worse  than 
nothing. 

54.  In  this  way,  it  frequently  happens,  in  the  course  of 
an  algebraic  process,  that  a  negative  quantity  is  brought  to 
stand  alone.  It  has  the  sign  of  subtraction,  without  being 
connected  with  any  other  quantity,  from  which  it  is  to  be 
subtracted.  This  denotes  that  a  previous  subtraction  has  left 
a  remainder,  which  is  a  part  of  the  quantity  subtracted.  If 
the  latitude  of  a  ship  which  is  20  degrees  north  of  the  equa- 
tor, is  considered  positive,  and  if  she  sails  south  25  degrees ; 
her  motion  first  diminishes  her  latitude,  then  reduces  it  to 
nothing,  and  finally  gives  her  5  degrees  of  south  latitude. 
The  sign  —  prefixed  to  the  25  degrees,  is  retained  before  the 
5  to  show  that  this  is  what  remains  of  the  southward  motion, 
after  balancing  the  20  degrees  of  north  latitude.  If  the  mo- 
tion southward  is  only  15  degrees,  the  remainder  must  be  +5, 
instead  of  —5,  to  show  that  it  is  a  part  of  the  ship's  northern 
latitude,  which  has  been  thus  far  diminished,  but  not  reduced 
to  nothing.  The  balance  of  a  book  account  will  be  positive 
or  negative,  according  as  the  debt  or  the  credit  is  the  greater 
of  the  two.  To  determine  to  which  side  the  remainder 
belongs,  the  sign  must  be  retained,  though  there  is  no  other 
quantity,  from  which  this  is  again  to  be  subtracted,  or  to 
which  it  is  to  be  added. 

55.  When  a  quantity  continually  decreasing  is  reduced 
to  nothing,  it  is  sometimes  said  to  become  afterwards  less 
than   nothing.      But   this   is   an   exceptionable   manner  of 


AXIOMS.  21 

speaking.*  No  quantity  can  be  really  less  than  nothing.  It 
may  be  diminished,  till  it  vanishes,  and  gives  place  to  an  op- 
posite quantity.  The  latitude  of  a  ship  crossing  the  equator, 
is  first  made  less,  then  nothing,  and  afterwards  contrary  to 
what  ^it  was  before.  The  north  and  south  latitudes  may 
therefore  be  properly  distinguished,  by  the  signs  +  and  —  ; 
all  the  positive  degrees  being  on  one  side  of  0,  and  all  the 
negative,  on  the  other ;  thus, 
+6,  +5,  +4,  +3,  +2,  +1,  0,  -1,  -2,  -3,  -4,  -5,  &c. 

The  numbers  belonging  to  any  other  series  of  opposite  quan- 
tities, may  be  arranged  in  a  similar  manner.  So  that  0  may 
be  conceived  to  be  a  kind  of  dividing  point  between  positive 
and  negative  numbers.  On  a  thermometer,  the  degrees  above 
0  may  be  considered  positive,  and  those  below  0,  negative. 

56.  A  quantity  is  sometimes  said  to  be  subtracted  from  0. 
By  this  is  meant,  that  it  belongs  on  the  negative  side  of  0. 
But  a  quantity  is  said  to  be  added  to  0,  when  it  belongs  on 
the  positive  side.  Thus,  in  speaking  of  the  degrees  of  a 
thermometer,  0+6  means  6  degrees  above  0;  and  0—6,  6 
degrees  below  0. 

Axioms. 

*>7.  The  object  of  mathematical  inquiry  is,  generally,  to 
investigate  some  unknown  quantity,  and  discover  how  great 
it  is.  This  is  effected,  by  comparing  it  with  some  other 
quantity  or  quantities  already  known.  .The  dimensions  of  a 
stick  of  timber,  are  found,  by  applying  to  it  a  measuring  rule 
of  known  length.  The  weight  of  a  body  is  ascertained,  by 
placing  it  in  one  scale  of  a  balance,  and  observing  how  many 
pounds  in  the  opposite  scale,  will  equal  it.  And  any  quantity 
is  determined,  when  it  is  found  to  be  equal  to  some  known 
quantity  or  quantities. 

Let  a  and  b  be  known  quantities,  and  y,  one  which  is  un- 
known. Then  y  will  become  known,  if  it  be  discovered  to 
be  equal  to  the  sum  of  a  and  b ;  that  is,  if 

y=a-\-b. 

*  The  expression  "  less  than  nothing,"  may  not  be  wholly  improper ;  if  it  is 
intended  to  be  understood,  not  literally,  but  merely  as  a  convenient  phrase 
adopted  for  the  sake  of  avoiding  a  tedious  circumlocution ;  as  we  say  "  the  sun 
rises,"  instead  of  saying  "the  earth  rolls  round,  and  brings  the  sun  into  view." 
The  use  of  it  in  this  manner,  is  warranted  by  Newton,  Euler  and  others. 


22  AXIOMS. 

An  expression  like  this,  representing  the  equality  between 
one  quantity  or  set  of  quantities,  and  another,  is  called  an 
equation.  It  will  be  seen  hereafter,  that  much  of  the  business 
of  algebra  consists  in  finding  equations,  in  which  some  un- 
known quantity  is  shown  to  be  equal  to  others  which  are 
known.  But  it  is  not  often  the  fact,  that  the  first  comparison 
of  the  quantities,  furnishes  the  equation  required.  It  will 
generally  be  necessary  to  make  a  number  of  additions,  sub- 
tractions, multiplications,  &c.  before  the  unknown  quantity 
is  discovered.  But  in  all  these  changes,  a  constant  equality 
must  be  preserved,  between  the  two  sets  of  quantities  com- 
pared. This  will  be  done,  if,  in  making  the  alterations,  we 
are  guided  by  the  following  axioms.  These  are  not  inserted 
here,  for  the  purpose  of  being  proved  ;  for  they  are  self-evident. 
(Art.  7.)  But  as  they  must  be  continually  introduced  or 
implied,  in  demonstrations  and  the  solutions  of  problems, 
they  are  placed  together,  for  the  convenience  of  reference. 

58.  Axiom  1.  If  the  same  quantity  or  equal  quantities  be 
added  to  equal  quantities,  their  sums  will  be  equal. 

2.  If  the  same  quantity  or  equal  quantities  be  subtracted 
from  equal  quantities,  the  remainders  will  be  equal. 

3.  If  equal  quantities  be  multiplied  into  the  same,  or  equal 
quantities,  the  products  will  be  equal. 

4.  If  equal  quantities  be  divided  by  the  same,  or  equal  quan- 
tities, the  quotients  will  be  equal. 

5.  If  the  same  quantity  be  both  added  to  and  subtracted 
from  another,  the  value  of  the  latter  will  not  be  altered. 

6.  If  a  quantity  be  both  multiplied  and  divided  by  another, 
the  value  of  the  former  will  not  be  altered. 

7.  If  to  unequal  quantities,  equals  be  added,  the  greater 
will  give  the  greater  sum. 

8.  If  from  unequal  quantities,  equals  be  subtracted,  the 
greater  will  give  the  greater  remainder. 

9.  If  unequal  quantities  be  multiplied  by  equals,  the  greater 
will  give  the  greater  product. 

10.  If  unequal  quantities  be  divided  by  equals,  the  greater 
will  give  the  greater  quotient. 

11.  Quantities  which  are  respectively  equal  to  any  other 
quantity  are  equal  to  each  other. 

12.  The  whole  of  a  quantity  is  greater  than  a  part. 


AXIOMS.  23 

This  is,  by  no  means,  a  complete  list  of  the  self-evident 
propositions,  which  are  furnished  by  the  mathematics.  It  is 
not  necessary  to  enumerate  them  all.  Those  have  been 
selected,  to  which  we  shall  have  the  most  frequent  occasion 
to  refer. 

59.  The  investigations  in  algebra  are  carried  on,  princi- 
pally, by  means  of  a  series  of  equations  and  proportions.  But 
instead  of  entering  directly  upon  these,  it  will  be  necessary 
to  attend  in  the  first  place,  to  a  number  of  processes,  on  which 
the  management  of  equations  and  proportions  depends. 
These  preparatory  operations  are  similar  to  the  calculations 
under  the  common  rules  of  arithmetic.  We  have  addition, 
multiplication,  division,  involution,  &c.  in  algebra,  as  well  as 
in  arithmetic.  But  this  application  of  a  common  name,  to 
operations  in  these  two  branches  of  the  mathematics,  is  often 
the  occasion  of  perplexity  and  mistake.  The  learner  natu- 
rally expects  to  find  addition  in  algebra  the  same  as  addition 
in  arithmetic.  They  are  in  fact  the  same,  in  many  respects : 
in  all  respects  perhaps,  in  which  the  steps  of  the  one  will 
admit  of  a  direct  comparison,  with  those  of  the  other.  But 
addition  in  algebra  is  more  extensive,  than  in  arithmetic. 
The  same  observation  may  be  made  concerning  several  other 
operations  in  algebra.  They  are,  in  many  points  of  view, 
the  same  as  those  which  bear  the  same  names  in  arithmetic. 
But  they  are  frequently  extended  farther,  and  comprehend 
processes  which  are  unknown  to  arithmetic.  This  is  com- 
monly owing  to  the  introduction  of  negative  quantities. 
The  management  of  these  requires  steps'  which  are  unneces- 
sary, where  quantities  of  one  class  only  are  concerned.  It 
will  be  important,  therefore,  as  we  pass  along,  to  mark  the 
difference  as  well  as  the  resemblance,  between  arithmetic  and 
algebra ;  and,  in  some  instances,  to  give  a  new  definition, 
accommodated  to  the  latter. 


24  ADDITION. 

SECTION    II. 

ADDITION. 

Art.  60.  Addition  is  the  collecting  of  several  algebraic 
expressions  into  one,  so  as  to  represent  their  aggregate  value. 
This  may  always  be  done  by  the  following 

Rule. 

Write  the  quantities  to  be  added,  one  after  another,  with 
their  signs ;  observing  that  a  quantity  to  which  no  sign  is 
prefixed,  is  to  be  considered  positive. 

6 1 .  Thus  a  added  to  b,  is  evidently,  according  to  the  alge- 
braic notation,  a+b.  And  a  added  to  the  sum  of  b  and  c,  is 
a+b+c.  And  a+b,  added  to  c-f-rf,  is  a+b+c+d.  In  the 
same  manner,  if  the  sum  of  any  quantities  whatever,  be 
added  to  the  sum  of  any  others,  the  expression  for  the  whole, 
will  contain  all  these  quantities  connected  by  the  sign  +. 

62.  Again,  if  the  difference  of  a  and  b  be  added  to  c ;  the 
sum  will  be  a— b  added  to  c,  that  is  a  —  b-\-c.  And  if  a—b 
be  added  to  c—d,  the  sum  will  be  a— b+c— d.  In  one  of  the 
compound  quantities  added  here,  a  is  to  be  diminished  by  b, 
and  in  the  other,  c  is  to  be  diminished  by  d\  the  sum  of  a 
and  c  must  therefore  be  diminished,  both  by  b,  and  by  d,  that 
is,  the  expression  for  the  sum  total,  must  contain  —b  and 
—d.  On  the  same  principle,  all  the  quantities  which,  in  the 
parts  to  be  added,  have  the  negative  sign,  must  retain  this 
sign  in  the  amount.  Thus  a+2b—c,  added  to  d—h  —  m,  is 
a-\-2b—c+d—h—m. 

63.  The  sign  must  be  retained  also,  when  a  positive  quan- 
tity is  to  be  added,  to  a  single  negative  quantity.  If  a  be 
added  to  —b,  the  sum  will  be  —b+a.  Here  it  may  be  ob- 
jected, that  the  negative  sign  prefixed  to  b,  shows  that  it  is 
to  be  subtracted.  What  propriety  then  can  there  be  in  add- 
ing the  two  quantities  ?  In  reply  to  this,  it  may  be  observed, 
that  the  sign  prefixed  to  b  while  standing  alone,  signifies  that 


REDUCTION      OF     LIKE      TERMS.  25 

b  is  to  be  subtracted,  not  from  a,  but  from  some  other  quan- 
tity, which  is  not  here  expressed.  Thus  —  b  may  represent 
the  loss,  which  is  to  be  subtracted  from  the  stock  in  trade. 
(Art.  50.)  The  object  of  the  calculation,  however,  may  not 
require  that  the  value  of  this  stock  should  be  specified.  But 
the  loss  is  to  be  connected  with  a  profit  on  some  article. 
Suppose  the  profit  is  2000  dollars,  and  the  loss  400.  The 
inquiry  then  is,  what  is  the  value  of  2000  dollars  profit,  when 
connected  with  400  dollars  loss  ? 

The  answer  is  evidently  2000—400,  which  shows  that  2000 
dollars  are  to  be  added  to  the  stock,  and  400  subtracted  from 
it ;  or  which  will  amount  to  the  same,  that  the  difference  be- 
tween 2000  and  400  is  to  be  added  to  the  stock. 

These  instances  are  sufficient  to  show  that  the  preceding 
rule  is  adapted  to  every  case  in  addition. 

64.  It  is  immaterial,  in  adding,  in  what  order  the  terms 
are  arranged.  The  sum  of  a  and  b  and  c  is  either  a-\-b+ c, 
or  a+c+b,  or  c-\-b+a.  For  it  evidently  makes  no  differ- 
ence, which  of  the  quantities  is  added  first.  The  sum  of  6 
and  3  and  9,  is  the  same  as  3  and  9  and  6,  or  9  and  6  and  3. 

And  a+m—n,  is  the  same  as  a  —  n+m.  For  it  is  plainly 
of  no  consequence,  whether  we  first  add  m  to  a,  and  after- 
wards subtract  n ;  or  first  subtract  n,  and  then  add  m. 

65.  It  is  to  be  observed  that  addition  in  algebra  does  not 
always  signify  augmentation,  as  it  does  in  arithmetic.  The 
algebraic  sum  of  two  quantities  which  have  opposite  signs, 
may  be  less  than  either  of  them.  Thus  the  aggregate  value 
of  7  and  —4,  is  not  11  but  3. 


Reduction  of  Like  Terms. 

66.  Though  connecting  quantities  by  their  signs  is  all 
which  is  essential  to  addition  ;  yet  it  is  desirable  to  make  the 
expression  as  simple  as  may  be,  by  reducing  several  terms  to 
one.     The  amount  of  Sa,  and  6b,  and  4a,  and  5b,  is 
3a+Gb+4a+5b. 

But  this  may  be  abridged.  The  first  and  third  terms  may  be 
brought  into  one ;  and  so  may  the  second  and  fourth.  For 
3  times  a,  and  4  times  a,  make  7  times  a.  And  6  times  b, 
and  5  times  b,  make  11  times  b.  The  sum  when  reduced  is 
therefore  la+llb. 

3 


26  ADDITION. 

For  making  the  reductions  connected  with  addition,  two 
rules  are  given,  adapted  to  the  two  cases,  in  one  of  which, 
the  quantities  and  signs  are  alike,  and  in  the  other,  the  quan- 
tities are  alike,  but  the  signs  are  unlike.  Like  quantities  are 
the  same  powers  of  the  same  letters.     (Art.  39.) 

Case  I. 

67.  To  reduce  several  terms  to  one  when  the  quantities 
are  alike,  and  the  signs  alike, 

Add  the  co-efficients,  annex  the  common  letter  or  letters,  and 
prefix  the  common  sign. 

Thus  to  reduce  Sb+7b,  that  is  +Sb+7b  to  one  term,  add 
the  co-efficients  3  and  7 ;  to  the  sum  10,  annex  the  common 
letter  b,  and  prefix  the  sign  +.  The  expression  will  then  be 
-\-\0b.  That  3  times  any  quantity,  and  7  times  the  same 
quantity,  make  10  times  that  quantity,  needs  no  proof. 

Examples. 

be  2hx  lb  +     xy  4nxy+  ar  cdxy+  Sgm 

2bc  hx  8b  +  Sxy  lnxy+3ar  2cdxy-\-     gm 

9bc  5hx  2b  +  2xy  2nxy+5ar  5cdxy+  Igm 

Sbc  Shx  6b  +  5xy  nxy-\-  ar  7cdxy+8gm 


I5bc  23b+llxy  15cdxy+l9gm 

The  mode  of  proceeding  will  be  the  same,  if  the  signs  are 
negative. 

Thus  —  Sbc—bc—  bbc,  becomes,  when  reduced,  —  9bc. 

And  —  ax— Sax— 2ax=— 6ax. 

Or  thus, 

— Sb2c         —7xy2       —  2ab—     my  —  b2h—  acnx 

—  b2c         —  xy3        —     ab—  Smy  —4b2h—9acnx 

—  5b2c         —Sxy3        —  lab—  Smy  —  b2h—2acnx 


—9b2c  —10ab—l2my 

68.  It  may  perhaps  be  asked  here,  as  in  Art.  63,  what  pro- 
priety there  is,  in  adding  quantities,  to  which  the  negative 


REDUCTION      OF     LIKE      TERMS.  27 

sign  is  prefixed ;  a  sign  which  denotes  subtraction  ?  The  an- 
swer to  this  is,  that  when  the  negative  sign  is  applied  to  sev- 
eral quantities,  it  is  intended  to  indicate  that  these  quantities 
are  to  be  subtracted,  not  from  one  another,  but  from  some 
other  quantity  marked  with  the  contrary  sign.  Suppose  that, 
in  estimating  a  man's  property,  the  sum  of  money  in  his  pos- 
session is  marked  +,  and  the  debts  which  he  owes  are  mark- 
ed — .  If  these  debts  are  200,  300,  500  and  700  dollars,  and 
if  a  is  put  for  100  ;  they  will  together  be  —2a— 3a— 5a— la. 
And  the  several  terms  reduced  to  one,  will  evidently  be  —17a, 
that  is,  1700  dollars. 

69.  Before  attending  to  the  second  case,  it  is  to  be  ob- 
served that  two  terms  may  be  reduced  to  one,  when  the  quan- 
tities are  alike,  but  the  signs  unlike,  by  the  following  rule. 

Take  the  less  co-efficient  from  the  greater,  to  the  difference 
annex  the  common  letter  or  letters,  and  prefix  the  sign  of  the 
greater  co-efficient. 

Thus,  instead  of  8a— 6a,  we  may  write  2a. 
And  instead  of  lb— 2b,  we  may  put  5b. 
For  the  simple  expression,  in  each  of  these  instances,  is 
equivalent  to  the  compound  one  for  which  it  is  substituted. 

To  +6b  —  8a*  5bc  —n*x  —dy+6m  —a—2b3y 
Add    —4b     +5a2     —Ibc        8n*x      4dy—  m      la—  b3y 


Sum   +26  —2bc  Sdy+5m 

70.  Here  again,  it  may  excite  surprise,  that  what  appears 
to  be  subtraction,  should  be  introduced  under  addition.  But 
this  subtraction  is  strictly  speaking,  no  part  of  the  addition. 
It  belongs  to  a  consequent  reduction.  Suppose  6b  is  to  be 
added  to  a— 4b.  The  sum  is  a— 4b+6b.  (Art.  60.)  But 
this  expression  may  be  rendered  more  simple.  As  it  now 
stands,  4b  is  to  be  subtracted  from  a,  and  6b  added.  But  the 
amount  will  be  the  same,  if,  without  subtracting  any  thing,  we 
add  2b,  making  the  whole  a-\-2b.  And  in  all  similar  instances, 
the  balance  of  two  quantities,  may  be  substituted  for  the  quan- 
tities themselves. 

If  two  equal  quantities  have  contrary  signs,  they  destroy 
each  other,  and  may  be  cancelled.  Thus  +6b—6b=0:  And 
3X6-18=0:  And  lbc-lbc=0. 


28  ADDITION. 


Case  II. 


7 1 .  To  reduce  several  terms  to  one,  when  the  quantities 
are  alike,  but  have  different  signs, 

Find  the  difference  between  the  sum  of  the  positive  and  the 
sum  of  the  negative  co-efficients ;  to  this  prefix  the  sign  of 
the  greater  sum,  and  annex  the  common  letter  or  letters. 

Ex.  1.  Reduce  13b+Gb+b— 4b— 5b— lb,  to  one  term. 

By  Art.  67,  I3b+6b+  b=     20b 
And  -4b-5b-lb=-l6b 


By  Art.  69,  20b—  I6b=  4b,   which  is  the  value 

of  all  the  given  quantities,  taken  together. 
Ex.  2.  Reduce  3xy— xy+2xy— lxy+4xy— 9xy— 6xy+  Ixy. 

The  positive  terms  are    Sxy  The  negative  terms  are  —  xy 

2xy  —  Ixy 

4xy  —  9xy 

Ixy  —  6xy 

And  their  sum  is  I6xy  —23xy 

Then     16xy—23xy=—lxy. 
Ex.  3.  c2z-5c2x+2c2x-lc2x~c2x+9c2x+4c2x-3c2x=0. 

4.  5h3xy—2h3xy+h3xy+9h3xy—4h3xy= 

5.  —abch—6abch+7abch—8abch-\-2abch= 

73.  If  the  letters,  in  the  several  terms  to  be  added,  are 
different,  they  can  only  be  placed  after  each  other,  with  their 

f roper  signs.     They  can  not  be  united  in  one  simple  term, 
f  4b,  and  —  6y,  and  3x,  and  11  h,  and  —5d,  and  6,  be  added, 
their  sum  will  be 

4b-6y+3x+llh-5d+6.  (Art.  60.) 
Different  letters  can  no  more  be  united  in  the  same  term, 
than  dollars  and  guineas  can  be  added,  so  as  to  make  a  sin- 
gle sum.  Six  guineas  and  four  dollars  are  neither  ten  guineas 
nor  ten  dollars.  Seven  hundred  and  five  dozen,  are  neither 
12  hundred  nor  12  dozen.     But  in  such  cases,  the  algebraic 


ADDITION.  29 

signs  serve  to  show  how  the  different  quantities  stand  related 
to  each  other ;  and  to  indicate  future  operations,  which  are 
to  be  performed,  whenever  the  letters  are  converted  into 
numbers.  In  the  expression  a +6,  the  two  terms  can  not  be 
united  in  one.  But  if  a  stands  for  15,  and  if,  in  the  course 
of  a  calculation,  this  number  is  restored ;  then  a-\-6  will  be- 
come 15+6,  which  is  equivalent  to  the  single  term  21.  In 
the  same  manner,  a— 6,  becomes  15—6,  which  is  equal  to  9. 
The  signs  keep  in  view  the  relations  of  the  quantities,  till  an 
opportunity  occurs  of  reducing  several  terms  to  one. 

73.  When  the  quantities  to  be  added  contain  several 
terms  which  are  alike,  and  several  which  are  unlike,  it  will 
be  convenient  to  arrange  them  in  such  a  manner,  that  the 
similar  terms  may  stand  one  under  another. 

To  3bc— 6d+2b— 3y  }     These  may  be  arranged  thus: 

Add  -3bc+x-3d+bg  >  3bc-6d+2b-3y+  x+bg+b 
And         2d+y+3x+b       J    —3bc—3d       +  y+3x 

2d 


The  sum  will  be     —7d+2b-2y+4x+bg+b 

Examples. 

1.  Add  7—xy,  to  ab+2,  and  3xy— 8A— 6. 

The  sum,  when  reduced,  is     3+2xy+ab— Qh. 

2.  Add  2y*-5x— 2ayy  to  bn— 6— 2y3  +  l. 

Ans.  bn—5—5x—2ay. 

3.  Add  bc+2x2—  axy,  to  9— x2—  h,  and  6h— 1— 4x2. 

4.  Add  5y2-9+ah+l,  to  12^-y3-10. 

5.  Add  bx2+4— y-3a3x,  to  9bx2  —  13+y+3a3x. 

6.  Add  5hn2+3ab—hn2+x,  to  hn2—x—7ab. 

7.  Add  4x2—  3bx— y,  to  x2  —  ay+bx— bn. 

8.  Add  ah+2mn— 3x,  to  4an— 2ah+4x. 


3* 


30  SUBTRACTION. 

SECTION    III. 

SUBTRACTION. 

Art.  74.  Subtraction  is  finding  the  difference  of  two 
quantities,  or  sets  of  quantities. 

Particular  rules  might  be  given,  for  the  several  cases  in 
subtraction.  But  it  is  more  convenient  to  have  one  general 
rule,  founded  on  the  principle,  that  taking  away  a  positive 
quantity  from  an  algebraic  expression,  is  the  same  in  effect, 
as  annexing  an  equal  negative  quantity ;  and  taking  away 
a  negative  quantity  is  the  same,  as  annexing  an  equal  posi- 
tive one. 

Suppose  +b  is  to  be  subtracted  from  a+b 

Taking  away  +b,  from  a+b,  leaves  a 

And  annexing  —b,  to  a-\-b>  gives  a+b— b 

But  by  axiom  5th,  a+b— b  is  equal  to  a 

That  is,  taking  away  a  positive  term,  from  an  algebraic 
expression,  is  the  same  in  effect,  as  annexing  an  equal  nega- 
tive term. 

Again,  suppose  —6  is  to  be  subtracted  from  a—b 

Taking  away  —6,  from  a— b,  leaves  a 

And  annexing  +b,  to  a— b,  gives  a —b+ b^ 

But  a—b+b  is  equal  to  a 

That  is,  taking  away  a  negative  term,  is  equivalent  to  an- 
nexing  a  positive  one. 

If  an  estate  is  encumbered  with  a  debt ;  to  cancel  this  debt 
is  to  add  so  much  to  the  value  of  the  estate.  Subtracting  an 
item  from  one  side  of  a  book  account,  will  produce  the  same 
alteration  in  the  balance,  as  adding  an  equal  sum  to  the 
opposite  side. 

So,  if  there  are  several  terms  in  the  subtrahend,  each  of 
them,  instead  of  being  taken  from  the  minuend,  may  be  an- 
nexed, with  its  sign  changed. 

To  place  this  in  another  point  of  view.  Suppose  that  c—d 
is  to  be  subtracted  from  a— b.     If  we  first  subtract  c,  the 


SUBTRACTION.  31 

remainder  will  be  a— 6— c,  which  is  too  small  by  d,  inasmuch 
as  we  were  to  subtract,  not  the  whole  of  c,  but  c  diminished 
by  d.  Adding  d  therefore,  we  obtain  a— 6— c+d  for  the  re- 
quired difference.  The  explanation  will  be  the  same,  if  there 
are  more  than  two  terms  in  the  subtrahend  or  minuend. 
Hence,  to  perform  subtraction  in  algebra,  we  have  the  fol- 
lowing general 

Rule. 

75.  Change  the  signs  of  all  the  quantities  to  be  subtracted, 
or  suppose  them  to  be  changed,  from  +  to  — ,  or  from  —  to  +, 
and  then  proceed  as  in  addition. 

This  rule  may  be  deduced  from  the  following  considera- 
tions, as  well  as  from  those  already  stated. 

If  two  quantities  be  increased  or  diminished  equally,  their 
difference  will  remain  unaltered.  Suppose  it  is  required  to 
find  the  difference  between  a+b,  and  c—d—h.  Before  seek- 
ing the  difference,  let  us  add  to  each  quantity,  the  expression 
— c+d+h,  which  consists  of  the  terms  of  the  subtrahend, 
with  their  signs  changed.  The  minuend  will  become  a+b—c 
+d-\-h.  And  the  subtrahend  will  become  c— d—  h— c+ d+h. 
This  reduced  becomes  nothing,  since  c  is  destroyed  by  —c, 
~d  by  +d,  and  —  A  by  +h.  (Art.  70.)  Now  as  the  subtrahend 
is  nothing,  the  remainder  must  be  the  same  as  the  minuend, 
namely,  a-\-b— c+d+h.  Thus  it  appears  that  the  required 
difference  is  obtained,  by  adding  to  the  original  minuend,  all 
the  terms  of  the  subtrahend  with  their  signs  changed.  And 
hence  we  derive  the  general  rule  for  subtraction,  as  stated 
above. 

Although  this  rule  is  adapted  to  every  case  of  subtraction  ; 
yet  there  may  be  an  advantage  in  giving  some  of  the  exam- 
ples in  distinct  classes. 

76.  In  the  first  place,  the  signs  may  be  alike,  and  the 
minuend  greater  than  the  subtrahend. 

From  +28       166       Uda       -28       -166       -Uda 

Subtract       +16       126         6da       —16       —126       —  6da 

Difference     +12        46         8da       —12       —  46       —  Sda 

Here,  in  the  first  example,  the  +  before  16  is  supposed  to 
be  changed  into  — ,  and  then,  the  signs  being  unlike,  the  two 


32  SUBTRACTION. 

terms  are  brought  into  one,  by  reduction,  as  in  addition. 
(Art.  69.)  The  two  next  examples  are  subtracted  in  the 
same  way.  In  the  three  last,  the  —  in  the  subtrahend,  is 
supposed  to  be  changed  into  +. 

It  may  be  well  for  the  learner,  at  first,  to  write  out  the  ex- 
amples ;  and  actually  to  change  the  signs,  instead  of  merely 
conceiving  them  to  be  changed.  But  when  he  has  become 
familiar  with  the  operation,  the  latter  method  is  much  to  be 
preferred. 

This  case  is  the  same  as  subtraction  in  arithmetic.  The 
two  next  cases  do  not  occur  in  common  arithmetic. 

77.  In  the  second  place,  the  signs  may  be  alike,  and  the 
minuend  less  than  the  subtrahend. 

From  +16         125        6da       -16       -126       -  6da 

Subtract       +28         166       Uda       -28       -166       -Uda 


Difference    -12       -46     Sda       +12  46  8da 

The  same  quantities  are  given  here,  as  in  the  preceding 
article,  for  the  purpose  of  comparing  them  together.  But  the 
minuend  and  subtrahend  are  made  to  change  places.  The 
mode  of  subtracting  is  the  same.  In  this  class,  a  greater 
quantity  is  taken  from  a  less ;  in  the  preceding,  a  less  from  a 
greater.  By  comparing  them,  it  will  be  seen,  that  there  is  no 
difference  in  the  answers,  except  that  the  signs  are  opposite. 
Thus  166—126  is  the  same  as  126—166,  except  that  one  is 
+46,  and  the  other  —46.  That  is,  a  greater  quantity  sub- 
tracted from  a  less,  gives  the  same  result,  as  the  less  sub- 
tracted from  the  greater,  except  that  the  one  is  positive,  and 
the  other  negative.     See  Art.  53  and  54. 

78.  In  the  third  place,  the  signs  may  be  unlike. 
From      +28       +166       +Uda       -28       -166       -Uda 
Sub.        -16       -126       -  6da       +16       +126       +  Gda 


Dif.         +44  286  20da       -44       -286       -20da 

From  these  examples,  it  will  be  seen  that  subtraction  in 
algebra  does  not  always  imply  diminution,  as  it  does  in  arith- 
metic. The  difference  between  a  positive  and  a  negative 
quantity,  is  greater  than  either  of  the  two  quantities.  In  a 
thermometer,  the  difference  between  28  degrees  above  cipher, 
and  16  below,  is  44  degrees.     The  difference  between  gain- 


SUBTRACTION. 


ing  1000  dollars  in  trade,  and  losing  500,  is  equivalent  to 
1500  dollars. 

79.  Subtraction  may  be  proved,  as  in  arithmetic,  by  add- 
ing the  remainder  to  the  subtrahend.  The  sum  ought  to  be 
equal  to  the  minuend,  upon  the .  obvious  principle,  that  the 
difference  of  two  quantities  added  to  one  of  them,  is  equal  to 
the  other.  This  serves  not  only  to  correct  any  particular 
error,  but  to  verify  the  general  rule. 

From  2xy—  1  3a2-\-xy  hy—  ah 

Sub.  —  xy+1  5a2+xy         5hy—6ah 


my- 
7my2- 


ny 
5ny 


Dif.  3xy-8 

From     3abm—  xy 
Sub.  —labm+Sxy 


•  l—3xy3 
■l5+2xy* 


—4hy+5ah 

ax+  lb     -2n*+3bh 
-4ax+15b       +n*+3bh 


5ax—  8b 


Rem.   lOabm—lxy 

80.  When  there  are  several  terms  alike,  they  may  be  re- 
duced as  in  addition. 

1.  From    ab,  subtract  3am+am+7am+2am+6am. 

Ans.  ab—3am—am—lam—2am—6am=ab—19am.  (Art.  67.) 

2.  From    y,  subtract  —a— a— a— a. 

Ans.  y+a+a+a+a=y+4a. 

3.  From  ax— bc+3ax+lbc,  subtract  4bc—2ax+bc+4ax. 

Ans.  ax—bc-\-3ax+lbc—4bc+2ax—bc—4ax 
=2ax+bc.     (Art.  71.) 

4.  From     3x2  —  2ay3+bc,    subtract    x2  —  4bc+ay3— 3x2. 

8 1  •  When  the  letters  in  the  minuend  are  different  from 
those  in  the  subtrahend,  the  latter  are  subtracted,  by  first 
changing  the  signs,  and  then  placing  the  several  *terms  one 
after  another,  as  in  addition.     (Art.  72.) 

From     3a&+8— my-\-dh,    subtract    x—dr+4hy—bmx. 
Ans.  3ab+8—my+dh—x+dr—4hy+bmx. 

82.  The  sign  — ,  placed  before  the  marks  of  parenthesis, 
which  include  a  number  of  quantities,  indicates  that  the  in- 
cluded polynomial  is  to  be  subtracted,  and  therefore  requires, 
that  when  those  marks  are  removed,  the  signs  of  all  the 
terms  of  the  polynomial  should  be  changed. 


34  MULTIPLICATION. 

Thus  a—(b-~c+d)  signifies  that  the  quantities  b,  —c  and 
+d,  are  to  be  subtracted  from  a.  The  expression  will  then 
become  a—b+c—d. 

2.  13ad+xy+d-  (7ad-xy+d+hm-ry)=6ad+2xy-hm+ry. 

3.  7abc-8+7x-(3abc-8-dx+r)=4abc+7x+dx-r. 

4.  5x—y2+hn—(2hn—4y2—x-\~ah+6x—ny)=z 

5.  2x*+ay-12-(l-x3+2h-13-3ay-c)  = 

6.  ab— 3ch—2—  (3— d—  5ab— ch+y)  = 

83.  On  the  other  hand,  when  a  number  of  quantities  are 
introduced  within  the  marks  of  parenthesis,  with  —  immedi- 
ately preceding ;  the  signs  must  be  changed. 

Thus     -m+b-dx+3h=-(m-b+dx-3h). 

Polynomials  may  accordingly  be  written  in  various  ways. 
For  example,  x—ab+9—y*—7h,  may  be  changed  to 
x-(ab-9+y'+7h),  or  x-ab-(-9+y2+7h),  or  x-ab 
+9— (y*+7*),  or  x-(ab-9)-(y2+7h);  each  of  these 
expressions  being  equivalent  to  the  first. 


SECTION    IV. 


MULTIPLICATION. 


Art.  84.  In  addition,  one  quantity  is  connected  with  an- 
other. It  is  frequently  the  case,  that  the  quantities  brought 
together  are  equal;  that  is,  a  quantity  is  added  to  itself. 

As     a+a=2a  a+a+a+a—4a 

a+a+a=3a  a+a+a+a+a=5a,  &c. 

This  repeated  addition  of  a  quantity  to  itself,  is  what  was 
originally  called  multiplication.  But  the  term,  as  it  is  now 
used,  has  a  more  extensive  signification.     We  have  frequent 


MULTIPLICATION.  35 

occasion  to  repeat,  not  only  the  whole  of  a  quantity,  but  a 
certain  portion  of  it.  If  the  stock  of  an  incorporated  com- 
pany is  divided  into  shares,  one  man  may  own  ten  of  them, 
another  five,  and  another  a  part  only  of  a  share,  say  two- 
fifths.  When  a  dividend  is  made,  of  a  certain  sum  on  a 
share,  the  first  is  entitled  to  ten  times  this  sum,  the  second  to 
jive  times,  and  the  third  to  only  two-fifths  of  it.  As  the  ap- 
portioning of  the  dividend,  in  each  of  these  instances,  is  upon 
the  same  principle,  it  is  called  multiplication  in  the  last,  as 
well  as  in  the  two  first. 

According  to  this  view  of  the  subject ; 

8*>.  Multiplying  by  a  whole  number  is  taking  the  multi- 
plicand as  many  times,  as  there  are  units  in  the  multiplier. 

Multiplying  by  1,  is  taking  the  multiplicand  once,  as  a. 
Multiplying  by  2,  is  taking  the  multiplicand  twice,  as  a+a. 
Multiplying  by  3,  is  taking  the  multiplicand   three  times,  as 
a+a-\-a,  &c. 

Multiplying  by  a  fraction  is  taking  a  certain  portion 
of  the  multiplicand  as  many  times,  as  there  are  like  portions 
of  a  unit  in  the  multiplier.* 

Multiplying  by  J,  is  taking  \  of  the  multiplicand,  once,  as  \a. 
Multiplying  by  f ,  is  taking  \  of  the  multiplicand,  twice,  as 

Multiplying  by  f ,  is  taking  }  of  the  multiplicand,  three  times. 

Hence,  if  the  multiplier  is  a  unit,  the  product  is  equal  to 
the  multiplicand :  If  the  multiplier  is  greater  than  a  unit, 
the  product  is  greater  than  the  multiplicand:  And  if  the 
multiplier  is  less  than  a  unit,  the  product  is  less  than  the 
multiplicand. 

86.  Every  multiplier  is  to  be  considered  a  number.  We 
sometimes  speak  of  multiplying  by  a  given  weight  or  meas- 
ure,  a  sum  of  money,  &c.  But  this  is  abbreviated  language. 
If  construed  literally,  it  is  absurd.  Multiplying  is  taking 
either  the  whole  or  a  part  of  a  quantity,  a  certain  number  of 
times.  To  say  that  one  quantity  is  repeated  as  many  times, 
as  another  is  heavy,  is  nonsense.  But  if  a  part  of  the  weight 
of  a  body  be  fixed  upon  as  a  unit,  a  quantity  may  be  multi- 

*  See  Note  B. 


36  MULTIPLICATION. 

plied  by  a  number  equal  to  the  number  of  these  parts  con- 
tained in  the  body.  If  a  diamond  is  sold  by  weight,  a  par- 
ticular price  may  be  agreed  upon  for  each  grain.  A  grain 
is  here  the  unit;  and  it  is  evident  that  the  value  of  the  dia- 
mond, is  equal  to  the  given  price  repeated  as  many  times,  as 
there  are  grains  in  the  whole  weight.  We  say  concisely,  that 
the  price  is  multiplied  by  the  weight ;  meaning  that  it  is  mul- 
tiplied by  a  number  equal  to  the  number  of  grains  in  the 
weight.  In  a  similar  manner,  any  quantity  whatever  may 
be  supposed  to  be  made  up  of  parts,  each  being  considered  a 
unit,  and  any  number  of  these  may  become  a  multiplier. 

87.  As  multiplying  is  taking  the  whole  or  a  part  of  a  quan- 
tity a  certain  number  of  times,  it  is  evident  that  the  product, 
must  be  of  the  same  nature  as  the  multiplicand. 

If  the  multiplicand  is  an  abstract  number,  the  product  will 
be  a  number. 

If  the  multiplicand  is  weight,  the  product  will  be  weight. 
If  the  multiplicand  is  a  line,  the  product  will  be  a  line.  Re- 
peating a  quantity  does  not  alter  its  nature.  It  is  frequently 
said,  that  the  product  of  two  lines  is  a  surface,  and  that  the 
product  of  three  lines  is  a  solid.  But  these  are  abbreviated 
expressions,  which  if  interpreted  literally  are  not  correct. 

88.  The  multiplication  of  fractions  will  be  the  subject 
of  a  future  section.  We  have  first  to  attend  to  multiplica- 
tion by  positive  whole  numbers.  And  there  are  here  two 
cases  to  be  considered ;  one,  in  which  the  factors  are  mo- 
nomials ;  and  the  other,  in  which  they  are  polynomials. 

Multiplication  of  Monomials. 

89.  Multiplying  by  a  whole  number,  as  above  defined 
(Art.  85.)  is  taking  the  multiplicand  as  many  times,  as  there 
are  units  in  the  multiplier.  Suppose  a  is  to  be  multiplied  by  b, 
and  that  b  stands  for  3.  There  are  then,  three  units  in  the 
multiplier  b.  The  multiplicand  must  therefore  be  taken  three 
times  ;  thus,  a+a+a=3a,  or  ba.     So  that, 

Multiplying  two  letters  together  is  nothing  more,  than 
writing  them  one  after  the  other,  either  with,  or  without  the 
sign  of  multiplication  between  them. 

Thus  b  multiplied  into  c  is  bXc,  or  be.  And  x  into  y,  is 
zXyy  or  x. y,  or  xy.     See  Art.  20. 


MULTIPLICATION.  37 

90.  If  more  than  two  letters  are  to  be  multiplied,  they 
must  be  connected  in  the  same  manner. 

Thus  a  into  b  and  c,  is  cba.  For  by  the  last  article,  a  into  b, 
is  ba.  This  product  is  now  to  be  multiplied  into  H  c 
stands  for  5,  then  ba  is  to  be  taken  five  times  thus* 

ba-\-ba+ba+ba+ba=5ba,  or  cba. 

The  same  explanation  may  be  applied  to  any  number  of  let- 
ters.    Thus,  am  into  xy,  is  amxy.    And  bh  into  mrx,  is  bhmrx. 

91.  It  is  immaterial  in  what  order  the  letters  are  arranged. 
The  product  ba  is  the  same  as  ab.  Three  times  five  is  equal 
to  five  times  three.  Let  the  number  5  be  represented  by  as 
many  points,  in  a  horizontal  line ;  and  the  number  3,  by  as 
many  points  in  a  perpendicular  line. 


Here  it  is  evident  that  the  whole  number  of  points  is  equal 
either  to  the  number  in  the  horizontal  row  three  times  repeat- 
ed, or  to  the  number  in  the  perpendicular  row  Jive  times  re- 
peated; that  is,  to  5X3,  or  3X5.  This  explanation  may  be 
extended  to  a  series  of  factors  consisting  of  any  numbers 
whatever.  For  the  product  of  two  of  the  factors  may  be 
considered  as  one  number.  This  may  be  placed  before  or 
after  a  third  factor ;  the  product  of  three,  before  or  after  a 
fourth;  &c. 

Thus     24=4X6  or  6X4  =4X3X2  or  4X2X3  or  2X3X4 

The  product  of  a,  b,  c  and  d,  is  abed,  or  acdby  or  dcbaf  or  bade* 

It  will  generally  be  convenient,  however,  to  place  the  let- 
ters in  alphabetical  order. 

92.  Powers  of  the  same  quantity  are  multiplied  by  add- 
ing  their  exponents. 

Thus  the  product  of  a2  into  a*  is  a5.  For  aa  is  the  same 
as  a2,  and  aaa  the  same  as  a3 ;  and  the  product  of  aa  into  aaa 
is  aaaaa,  that  is,  a5.     Also,  b2Xbi=b\  and  x2Xx10=x12. 

It  is  to  be  observed,  that  when  a  letter  has  no  exponent, 
one  is  to  be  understood.  Thus  Ax  A3,  is  the  same  as  h1  Xh3, 
that  is,  A4. 

4 


38  MULTIPLICATION. 

03.  When  the  letters  have  numerical  co- efficients,  these 
must  be  multiplied  together,  and  prefixed  to  the  product  of 
the  letters. 

Thus,  3a  into  2b,  is  6ab.  For  if  a  into  b  is  ab,  then  3  times 
a  into  b,  is  evidently  Sab ;  and  if,  instead  of  multiplying  by 
b,  we  multiply  by  twice  b,  the  product  must  be  twice  as  great ; 
that  is  2X3ab  or  6ab. 

Multiply     dab  5cd     3dy  lamh     Ibdh  h  x* 

Into  3xy  I2bh      my2  9by  x       I3h2x2 


Product    27 abxy  3dmy3  Ibdhx 

If  either  of  the  factors  consists  of  figures  only,  these  must  be 
multiplied  into  the  co-efficients  and  letters  of  the  other  factors. 

Thus  3ab  into  4,  is  \2ab.  And  36  into  2x,  is  12x.  And 
24  into  hy,  is  24%. 

94.  The  numeral  co-efficients  of  several  fellow-factors 
may  be  brought  together  by  multiplication. 

Thus  2aX3b  into  4aX5b  is  2aX3bX4aX5b,  or  120a3 b2 . 

For  the  co-efficients  are  factors  (Art.  31.),  and  it  is  imma- 
terial in  what  order  these  are  arranged.     (Art.  91.)    So  that 
2aX3bX4aX5b=2X3X4X5XaXaXbxb=120a2b2. 
The  product  of  3aX4bh  into  5mX6y,  is  360abhmy. 
The  product  of  4bX6d  into  x,  is  24bdx. 

Multiplication  of  Polynomials. 

95.  If  the  multiplicand  is  a  compound  quantity,  each  of 
its  terms  must  be  multiplied  into  the  multiplier. 

Thus  b+c+d  into  a,  is  ab-\-ac+ad.  For  the  whole  of  the 
multiplicand  is  to  be  taken  as  many  times,  as  there  are  units 
in  the  multiplier.  If  then  a,  stands  for  3,  the  repetitions  of 
the  multiplicand  are, 

b+c+d 

b+c+d 

b+c+d 


And  their  sum  is         3b+3c+3d,  that  is,  ab+ac+ad. 


MULTIPLICATION.  39 

Multiply        d+2xy         h2+2n      Shl+l  ax+2hy+5 

Into  3b  5hx  my  Sy 


Product     Sbd+6bxy  Shlmy+my 

96.  The  preceding  instances  must  not  be  confounded 
with  those  in  which  several  factors  are  connected  by  the 
sign  X ,  or  by  a  point.  In  the  latter  case,  the  multiplier  is 
to  be  written  before  the  other  factors  without  being  repeated. 

The  product  of  bxd  into  a,  is  abxd,  and  not  abXad.  For 
bxd  is  bd,  and  this  into  a,  is  abd.  (Art.  90.)  The  expression 
bxd  is  not  to  be  considered,  like  b+d,  a  compound  quantity 
consisting  of  two  terms.  Different  terms  are  always  sepa- 
rated by  +  or  — .  (Art.  38.)  The  product  of  bxhXmXy 
into  a,  is  aXbxhXmXy  or  abhmy.  But  b+h+m+y  into  a, 
is  ab+ah+am+ay.  The  product  of  3X5  into  4,  is  15X4  or 
60,  and  not  4X3X4X5,  which  is  240.  But  the  product  of 
3+5  into  4=4X3+4X5=12+20=32. 

97.  If  both  the  factors  are  compound  quantities,  each 
term  in  the  multiplier  must  be  multiplied  into  each  in  tlie 
multiplicand. 

Thus     a+b  into  c+d  is  ac+ad+bc+bd. 

For  the  units  in  the  multiplier  a+b  are  equal  to  the  units 
in  a  added  to  the  units  in  b.  Therefore  the  product  produ- 
ced by  a}  must  be  added  to  the  product  produced  by  b. 

The  product  of  c+d  into  a  is  ac+ad )       ,.  . 

The  product  of  c+d  into  b  is  bc+bd  >      J       '      ' 

The  product  oi  c+d  into  a+b  is  therefore  ac+ad+bc+bd. 

Mult.       Sx+d  Sbx+a  a+\ 

Into         2a+hm  2b2+3h        Sx+4 


Prod.     6ax+2ad+Shmx+dhm  Sax+Sx+^a+4: 

4.  Mult.     2h+7  into  6^+1.     Prod.  12dh+42d+2h+7. 

5.  Mult.     ax+b2+c  into  n+3+7a2.     Prod. 

6.  Mult.     l+x+2hy  into  5a+3b+6.     Prod. 


40  MULTIPLICATION. 

98.  When  two  or  more  terms  in  the  product  are  alike, 
it  will  be  expedient  to  set  one  under  the  other,  and  then  to 
unite  them,  by  the  rules  for  reduction  in  addition. 

Mult.       b+2a  b+c+2  x+6+a* 

Into         b+2a  i+c+3  2y+36+l 


b2+2ab  b2+bc+2b 

+2ab+4a2  be        +c2+2c 

+Sb         +3c+6 


Prod.       b2+4ab+4a2  b2  +2bc+5b+c2  +5c+6 

4.  Mult,     h+a+1  into3A+2a+5.     Prod. 

5.  Mult.     2a +3+xy  into  a+2+3xy.     Prod. 

6.  Mult.     c+3d+2e  into  bX2dXx.     Prod. 

99.  The  examples  in  multiplication  thus  far  have  been 
confined  to  positive  quantities.  But  in  algebra,  multiplica- 
tion is  performed  also  with  negative  quantities.  And  it  will 
now  be  necessary  to  consider  in  what  manner  the  result  will 
be  affected,  by  multiplying  positive  and  negative  quantities 
together.     We  shall  find, 

That  +  into  +  produces  + 

—  into  +  — 
+  into  —  — 

—  into  —  + 

All  these  may  be  comprised  in  one  general  rule,  which  it 
will  be  important  to  have  always  familiar. 

If  the  signs  of  the  factors  are  alike,  the  sign  of  the  pro- 
duct will  be  positive ;  but  if  the  signs  of  the  factors  are  un- 
like, the  sign  of  the  product  will  be  negative. 

100.  The  first  case,  that  of  +  into  +,  needs  no  farther 
illustration. 

The  second  is  —  into  +,  that  is,  the  multiplicand  is  nega- 
tive, and  the  multiplier  positive.  Here  —a  into  +4  is  —4a. 
For  the  repetitions  of  the  multiplicand  are,  -a-a-a-a=  -4a. 
And  —a  into  -\-b  is  —a  repeated  b  times,  that  is,  —ba. 


MULTIPLICATION.  41 

Mult.         b— 3a       5x—a  h—3d—4  y—a—3—2h 

Into        6y  b+2y        2y  2x-\-3b 


Prod.     6by-\8ay  2hy-6dy-8y 

101.  The  third  case  is  that  in  which  the  multiplicand  is 
positive,  but  the  multiplier  negative. 

The  effect  of  multiplying  by  a  negative  quantity  is  best 
shown  by  the  aid  of  a  compound  multiplier.  Thus,  it  is  easy 
to  ascertain  the  product  of  a  into  —4,  from  the  product  of  a  into 
the  binomial  6—4.  As  6— 4  is  equal  to  2,  the  product  will 
be  equal  to  2a.  This  is  less  than  the  product  of  6  into  a. 
To  obtain  then  the  product  of  the  compound  multiplier  6—4 
into  a,  we  must  subtract  the  product  of  the  negative  part, 
from  that  of  the  positive  part. 

Multiplying;         a  >  .     .  ( Multiplying    a 

T  „     *  >  is  the  same  as  <  T  A     r  J    &    -m 

Into  6-4)  {Into  2 

And  the  product  6a— 4a,  is  the  same  as  the  product    2a. 

Therefore  a  into  —4,  is  —4a.  And  this  must  be  the  pro- 
duct, as  well  when  —4  stands  alone,  as  when  it  forms  a  part 
of  a  compound  multiplier.  For  every  negative  quantity  must 
be  supposed  to  have  a  reference  to  some  other  which  is  posi- 
tive ;  though  the  two  may  not  always  stand  in  connection, 
when  the  multiplication  is  to  be  performed. 

If  the  multiplier  had  been  6+4  instead  of  6—4,  the  two 
products  6a  and  4a  must  have  been  added. 

Multiplying        a  >  .  C  Multiplying    a 

t  a.  A  .  .  i  is  tne  same  as  i  T  .  - «. 

Into  6+4  >  C  Into  10 

And  the  prod.     6a+4a   is  the  same  as  the  product  10a. 

This  shows  at  once  the  difference  between  multiplying  by 
a  positive  factor,  and  multiplying  by  a  negative  one.  In  the 
former  case,  the  sum  of  the  repetitions  of  the  multiplicand  is 
to  be  added  to  other  quantities,  and  in  the  latter,  subtracted 
from  them. 

Mult,     a+b  ^+3+2^2  3^+3 

Into        b— x  am  — be  ad— 6 


Prod.    ab+b2—ax—bx  3adh+3ad -18h-18 

4* 


42  MULTIPLICATION. 

1 02.  If  two  negatives  be  multiplied  together,  the  product 
will  be  positive;  — 4X—  a—Jt4a.  In  this  case,  as  in  the 
preceding,  the  repetitions  of  the  multiplicand  are  to  be  sub- 
tracted, because  the  multiplier  has  the  negative  sign.  These 
repetitions,  if  the  multiplicand  is  —a,  and  the  multiplier  —4, 
are  —  a— a— a— a=—4a.  But  this  is  to  be  subtracted  by 
changing  the  sign.     It  then  becomes  +4a. 

Suppose  —a  is  multiplied  into  (6—4).  As  6—4=2,  the 
product  is,  evidently,  twice  the  multiplicand,  that  is  —2a. 
But  if  we  multiply  —a  into  6  and  4  separately;  —a  into  6 
is  —6a,  and  —a  into  4  is  —4a.  (Art.  100.)  As  in  the  mul- 
tiplier, 4  is  to  be  subtracted  from  6 ;  so,  in  the  product,  —4a 
must  be  subtracted  from  —6a.  Now  —  4a  becomes  by  sub- 
traction +4a.  The  whole  product  then  is  —6a+4a,  which 
is  equal  to  —2a.     Or  thus, 

Multiplying        -«  J  isthesameas  5  Multiplying     -a 
Into  6—4  5  C  Into  2 


And  the  prod.  —6a+4a,  is  equal  to  the  product  —2a. 

It  is  often  considered  a  great  mystery,  that  the  product  of 
two  negatives  should  be  positive.  But  it  amounts  to 
nothing  more  than  this,  that  the  subtraction  of  a  negative 
quantity,  is  equivalent  to  the  addition  of  a  positive  one; 
(Art.  74.)  and  therefore,  that  the  repeated  subtraction  of  a 
negative  quantity,  is  equivalent  to  a  repeated  addition  of  a 
positive  one.  Taking  off  from  a  man's  hands  a  debt  of 
ten  dollars  every  month,  is  adding  ten  dollars  a  month  to  the 
value  of  his  property. 

Mult.       a— 4  a—2bx+h2     Say—b 

Into       3b— 6  3c— 5  6x—  1 


Prod.    Sab—  12b— 6a+24  18axy-6bx-3ay+b 

4.  Multiply     b2—  5— 2cx  into  1—b—cy. 

5.  Multiply     3ax+h— 2  into  by2—  n+1. 

103.  As  a  negative  multiplier  changes  the  sign  of  the 
quantity  which  it  multiplies ;  if  there  are  several  negative 
factors  to  be  multiplied  together, 


MULTIPLICATION.  43 

The  two  first  will  make  the  product  positive ; 
The  third  will  make  it  negative ; 
The  fourth  will  make  it  positive;  &c. 

Thus  —  a  X  —  b  =  +  ah       \  (two  factors. 

+abX—c=  —  abc       I  ,  \  three. 

_     ,    ,    ,     >the  product  of  <    - 
—  abcX—d=+abcd     j         r  |  /owr. 

+a&cdx  —e=—abcde  J  K^five. 

That  is,  the  product  of  any  even  number  of  negative  fac- 
tors is  positive ;  but  the  product  of  any  odd  number  of  nega- 
tive factors  is  negative. 

Thus  —  aX—  a=a2  And     —aX—aX—aX—a=a* 

—aX—aX—a=—a3  —aX—aX—aX—aX—a=—a5 

The  product  of  several  factors  which  are  all  positive,  is 
invariably  positive. 

104.  Positive  and  negative  terms  may  frequently  balance 
each  other,  so  as  to  disappear  in  the  product.  (Art.  70.)  A 
star  is  sometimes  put  in  the  place  of  a  deficient  term. 

Mult.  2a  -b  x+a2  a2+ab+b2 

Into  2a  +b  x—a2  a  —b 


4a2-2ab  a2+a2b+ab2 

+2ab-b2  -a2b-ab2-b* 


Prod.  4a2    *     -b2  a2       *       *     -63 

1 05.  For  many  purposes,  it  is  sufficient  merely  to  indi- 
cate the  multiplication  of  compound  quantities,  without  actu- 
ally multiplying  the  several  terms.     Thus  the  product  of 

a+b+c  into  h+m+y,  is  (a+b+c)  X  (h+m+y),   or  simply 
(a+b+c)  (h+m+y). 

The  product  of 

a+m  into  h+x  and  d+y,  is  (a+m)  (h+x)  (d+y). 

When  the  several  terms  are  multiplied  in  form,  the  expres- 
sion is  said  to  be  expanded.     Thus, 

(a+b)  (c+d)  becomes  when  expanded  ac+ad+bc+bd. 


44  MULTIPLICATION. 

106.  With  a  given  multiplicand,  the  less  the  multiplier, 
the  less  will  be  the  product.  If  then  the  multiplier  be  reduced 
to  nothing,  the  product  will  be  nothing.  Thus  aXO=0. 
And  if  0  be  one  of  any  number  of  fellow-factors,  the  product 
of  the  whole  will  be  nothing. 

Thus,  abXcX3dX0=3abcdX0=0. 
And     (a+b)  (c+d)  (A-m)xO=0. 

107.  Although,  for  the  sake  of  illustrating  the  different 
points  in  multiplication,  the  subject  has  been  drawn  out  into 
a  considerable  number  of  particulars ;  yet  it  will  scarcely  be 
necessary  for  the  learner,  after  he  has  become  familiar  with 
the  examples,  to  burden  his  memory  with  any  thing  more 
than  the  following  general  rule. 

Multiply  the  letters  and  co-efficients  of  each  term  in  the 
multiplicand,  into  the  letters  and  co-efficients  of  each  term  in 
the  multiplier ;  and  prefix  to  each  term  of  the  product,  the 
sign  required  by  the  principle,  that  like  signs  produce  +,  and 
different  signs  — . 

When  like  terms  occur  in  the  product,  they  are  to  be  uni- 
ted.    See  Art.  71  and  98. 

1.  Mult.  3a—x+2  into  a— 4x—  1. 

2.  Mult.  xyXSaX5  into  2n  +  l—3h*. 

3.  Mult.  5(x—3ab)  into  yX2X5hX3. 

4.  Mult.  3(2a2+bc-l)  into  d(6-2x-l). 

5.  Mult.  ax+5x— h—2  into  (c+d)(x— h). 

6.  Mult.  5x2y-(b-3c)  into  (b-1)  (n+1). 

7.  Mult.  ay-7+b(c-m)  into  —(2n-l+3x). 

Multiplication  by  Detached  Co-efficients. 

108.  There  are  certain  cases,  in  which  the  numeral  co- 
efficients may  be  employed,  apart  from  the  letters,  in  obtain- 
.ng  the  product  of  two  polynomials. 


MULTIPLICATION.  45 

Suppose  that  a2  +3ax+2x2  is  to  be  multiplied  by  2a* 
+5ax+x2.  The  operation  performed  in  the  usual  way,  is 
as  follows. 

a2+  3a  x+  2x2 
2a2  +  da  x+     x2 


2a*  +  6a3x+  4a2x2 

+  5a3x+15a2x2+10ax3 

+     a2x2+  3ax3+2x* 

2a*  +  lla3x+20a2x2  +  13ax3+2x* 

Now  with  regard  to  the  letters  and  exponents,  it  is  easy  to 
see,  before  multiplying,  that  as  each  term  in  the  multiplier 
and  each  in  the  multiplicand  is  of  the  second  degree,  each 
term  in  the  product  will  be  of  the  fourth  degree :  and  it  is 
obvious  that  the  first  term  will  contain  «4 ;  the  last,cc4 ;  and 
the  intervening  terms,  a3x,  a2x2 ,  ax3. 

We  may  then,  in  multiplying,  proceed  with  the  co-efficients 
alone,  as  if  they  were  accompanied  with  letters,  and  after 
having  thus  obtained  the  co-efficients  of  the  product,  annex 
to  them  their  proper  letters  and  exponents. 

This  process  is  exhibited  below. 

Co-efficients  of    a2+3ax+2x2,         1+3+2 
Co-efficients  of  2a2+5ax+  x2,        2+  5  +  1 


2+  6+4 

5+15+10 

1+  3+2 


Co-efficients  of  the  product,  2+11+20+13+2 


The  product  itself,  2a*  +  lla3x+20a2x2  +  l3ax3+2x* 

The  same  method  may  be  pursued,  when  any  of  the 
co-efficients  are  negative.  For  example,  the  product  of 
a2+2ax— 3x2  into  2a— x,  is  obtained  as  follows. 


46  MULTIPLICATION. 

1+2-3 
2-1 


2+4-6 
-1-2+3 

2+3-8+3 


The  product  sought,  2a3+3a2x-8ax3+3x3 

It  will  sometimes  happen  that  the  multiplier  and  multipli- 
cand are  of  the  same  form  as  in  the  preceding  examples,  ex- 
cept that  one  or  more  of  the  terms  is  wanting.  In  such 
cases,  0  may  be  put  in  the  place  of  the  co-efficient  of  each 
absent  term,  and  the  multiplication  be  then  performed  as 
above. 

Suppose  the  factors  are  a2  — 2&2,  a3—ab2  +3&3 ;  the  first  of 
which  is  incomplete  for  want  of  a  term  containing  ab ;  and 
the  last,  for  want  of  a  term  containing  a2  b. 

If  we  were  to  multiply  here  as  in  the  previous  cases,  the 
co-efficients  would  not  all  fall  in  their  proper  places,  and  we 
should  be  led  into  error.  But  this  difficulty  will  be  obviated, 
if  we  imagine  OXab  and  0Xa2b  to  stand  in  the  place  of  the 
absent  terms,  and  employ  the  co-efficient  0,  in  the  same  way 
with  other  co-efficients.  The  factors,  without  being  changed 
in  value,  will  thus  become, 

a2+0ab-2b2,  a3+0a2b-ab2 +3b*t 

and  the  multiplication  will  proceed  as  follows, 

Co-efficients  of  one  factor,         1+0—1+3 
Co-efficients  of  the  other,  1+0—2 


1+0-1+3 

-2-0+2-6 

Co-efficients  of  the  product,       1+0-3+3+2-6 


Product,  a5  -3a3b2  +3a2b*+2ab*  -66s 

There  is  no  term  here  containing  a* b,  because  the  corres- 
ponding co-efficient  is  0. 


MULTIPLICATION.  47 


Examples. 

1.  Multiply  i3  — b2x+bx2—  x3   by   b+x. 

2.  Multiply  y3— 3ya+3y—l    by    y2-l. 

3.  Multiply  xA+2x2y2+y*    by   x2— 2zy+ya. 

4.  Multiply  a2-l    by    a2  +  l. 

109.  The  following  theorems  relate  to  certain  cases  in 
Multiplication  of  frequent  occurrence,  and  should  be  care- 
fully learned. 

Theorem  I. 

The  square  of  the  Sum  of  two  quantities  is  equal  to  the 
square  of  the  first,  plus  twice  the  product  of  the  first  and 
second,  plus  the  square  of  the  second. 

This  may  be  expressed  algebraically  thus, 

(a+b)2=a2+2ab+b2, 

where  a  and  6  represent  the  two  quantities,  and  a +b  is  their 
sum.  In  proof  of  the  theorem,  it  is  sufficient  to  observe  that 
(a-\-b)2  is  the  same  as  (a-\-b)  (a+b) ;  which,  when  expanded, 
becomes  a2  +2ab+b2 . 

From  this,  the  square  of  the  sum  of  any  two  quantities 
may  be  at  once  obtained,  without  multiplication. 

Examples. 


1. 

(2a+xy)2  =4a2  +4axy+x2  y2 . 

5.     (90+5)*  = 

2. 

(b2+3y)2=b*+6b2y+9y2. 

6.     (5a2y+2ax2) 

3. 

(3x3  +  l)2  = 

7.     (4an2+7)2  = 

4. 

(100+1)2  = 

8.     (12  +  i)2  = 

Learners,  not  familiar  with  this  theorem,  are  apt  to  assume 
the  square  of  a+b  to  be  simply  a2-\-b2. 

Theorem  II. 

110.  The  square  of  the  Difference  of  two  quantities  is 
equal  to  the  square  of  the  first,  minus  twice  the  product  of 
the  first  and  second,  plus  the  square  of  the  second. 


48  MULTIPLICATION. 

For,  if  the  quantities  are  represented  by  a  and  b,  their 
difference  is  a— b ;  and  the  square  of  this  will,  by  multiplying, 
be  found  to  be  a2  —  2ab+b2  :  therefore,  (to  express  the  theo- 
rem algebraically,) 

(a-b)2=a2-2ab+b2. 
Examples. 


1. 

(a— 2x)2—a2  —  4ax+4x2. 

5. 

(l-2:r8)2  = 

2. 

(100- 1)2  =  10000-200+1, 

6. 

(3a2b-4ax2)2  = 

or  992  =9801. 

7. 

(20-TV)2  = 

3. 

(1000-3)2  = 

8. 

(2z2-i)2  = 

4. 

(5a;2-2y)2  = 

9. 

(    1     —      1      \2  — 

\i  o       Too/ 

Learners  often  assume  the  square  of  a— b  to  be  a2—  ft3, 
instead  of  a2—2ab-\~b2,  which  is  here  seen  to  be  the  true 
expression. 

Theorem  III. 

111.  The  product  of  the  sum  and  difference  of  two  quan- 
tities, is  equal  to  the  difference  of  their  squares. 

For,  if  the  quantities  are  a  and  b,  a+b  is  their  sum,  a— & 
their  difference;  and  the  product  of  a-\-b  into  a— 6,  as  will  be 
found  by  multiplying,  is  a2—b2. 

The  theorem  may  be  stated  algebraically  thus, 
(a+b)  (a-b)=a2-b2. 

Examples. 

1.  (2x+l)(2x-l)=4x2-l.  5.  (l+4a2)(l-4a2)  = 

2.  (3a+2x2)(3a-2x2)=9a2-4x*.  6.  (x2+7xy)  (x2  -7:ry)  = 

3.  (a2x+7z2y)  (a2x-7x2y)  =  7.  (10+^)  (10-yVH 

4.  (2z3y+l)(2a;3y-l)  =  8.  (100+7)  (100-7)=* 


DIVISION.  49 

SECTION    V. 

DIVISION. 

Art.  112.  In  multiplication,  we  have  two  factors  given, 
and  are  required  to  find  their  product.  By  multiplying  the 
factors  4  and  6,  we  obtain  the  product  24.  But  it  is  fre- 
quently necessary  to  reverse  this  process.  The  number  24, 
and  one  of  the  factors  may  be  given,  to  enable  us  to  find  the 
other.  The  operation  by  which  this  is  effected,  is  called 
Division.  We  obtain  the  number  4,  by  dividing  24  by  6. 
The  quantity  to  be  divided  is  called  the  dividend ;  the  given 
factor,  the  divisor ;  and  that  which  is  required,  the  quotient. 

113*  Division  is  finding  a  quotient,  which  multiplied 
into  the  divisor  will  produce  the  dividend* 

In  multiplication  the  multiplier  is  always  a  number. 
(Art.  86.)  And  the  product  is  a  quantity  of  the  same  kind,  as 
the  multiplicand.  (Art.  8  ,)  The  product  of  3  rods  into  4,  is 
12  rods.  When  we  come  to  division,  the  product  and  either 
of  the  factors  may  be  given,  to  find  the  other ;  that  is, 

The  divisor  may  be  a  number,  and  then  the  quotient  will 
be  a  quantity  of  the  same  kind  as  the  dividend ;  or, 

The  divisor  may  be  a  quantity  of  the  same  kind  as  the 
dividend ;  and  then  the  quotient  will  be  a  number. 

Thus  12  rods-r-3=4:  rods.     But  12  rods-r-S  ro<&=4. 
And  12  rods-r-24:=l  rod.      And  12  rods -f- 24  rods=±. 

In  the  first  case,  the  divisor  being  a  number,  shows  into 
how  many  parts  the  dividend  is  to  be  separated ;  and  the  quo- 
tient shows  what  these  parts  are. 

If  12  rods  be  divided  into  3  parts,  each  will  be  4  rods  long. 
And  if  12  rods  be  divided  into  24  parts,  each  will  be  half  a 
rod  long. 

In  the  other  case,  if  the  divisor  is  less  than  the  dividend, 
the  former  shows  into  what  parts  the  latter  is  to  be  divided ; 

*  The  remainder  is  here  supposed  to  be  included  in  the  quotient,  as  is  com- 
monly the  case  in  algebra. 

5 


50  DIVISION. 

and  the  quotient  shows  how  many  of  these  parts  are  contained 
in  the  dividend.  In  other  words,  division  in  this  case  con- 
sists in  rinding  how  often  one  quantity  is  contained  in  another. 

A  line  of  3  rods,  is  contained  in  one  of  12  rods,  four  times. 

But  if  the  divisor  is  greater  than  the  dividend,  and  yet  a 
quantity  of  the  same  kind,  the  quotient  shows  what  part  of 
the  divisor  is  equal  to  the  dividend. 

Thus  one  half  of  24  rods  is  equal  to  12  rods. 

Division  of  Monomials. 

114.  As  the  product  of  the  divisor  and  quotient  is  equal 
to  the  dividend,  the  quotient  may  be  found,  by  resolving  the 
dividend  into  two  such  factors,  that  one  of  them  shall  be  the 
divisor.     The  other  will,  of  course,  be  the  quotient. 

Suppose  abd  is  to  be  divided  by  a.  The  factors  a  and  bd 
will  produce  the  dividend.  The  first  of  these,  being  a  divisor, 
may  be  set  aside.     The  other  is  the  quotient.     Hence, 

When  the  divisor  is  found  as  a  factor  in  the  dividend,  the 
division  is  performed  by  cancelling  this  factor. 


Divid< 

3      CX 

dh 

d2rx 

hm2y 

dhxy3 

abscd 

abxy 

By 

C 

d 

d2r 

hm2 

dy3 

b5 

ax 

Quot.       xx  hx  by 

In  each  of  these  examples,  the  letters  which  are  common 
to  the  divisor  and  dividend,  are  set  aside,  and  the  other  letters 
form  the  quotient.  It  will  be  seen  at  once,  that  the  product 
of  the  quotient  and  divisor  is  equal  to  the  dividend. 

1 1 1*.  A  power  is  divided  by  another  power  of  the  same 
letter,  by  subtracting  the  exponent  of  the  divisor  from  that 
of  the  dividend. 

Thus  the  quotient  of  a5  divided  by  a2  is  a3.     For  a3  mul- 
tiplied into  a2  will  produce  a5.     See  Art.  92  and  113. 
Also,  b1-i-b=b(it  x10-r-x*=x2,  y6+y5=y. 

116.  In  performing  multiplication,  if  the  factors  contain 
numeral  figures,  these  are  multiplied  into  each  other.  (Art.  93.) 
Thus  Sa  into  lb  is  21  ab.     Now  if  this  process  is  to  be  re- 


DIVISION.  51 

versed,  it  is  evident  that  dividing  the  number  in  the  product, 
by  the  number  in  one  of  the  factors,  will  give  the  number  in 
the  other  factor.     The  quotient  of  2\ab-^3a  is  lb.     Hence, 

In  division,  if  there  are  numeral  co-efficients  prefixed  to 
the  letters,  the  co-efficient  of  the  dividend  must  be  divided,  by 
the  co-efficient  of  the  divisor. 

Divide  Qa2^  24h2x3y5  25dhr  20hn2  34drx  12my2 
By  2a2b3       Ghxy*  dh        5  34  y 

Quot.      3a  b*  25r  drx 

117.  In  division,  the  same  rule  is  to  be  observed  respect- 
ing the  signs,  as  in  multiplication  ;  that  is,  if  the  divisor  and 
dividend  are  both  positive,  or  both  negative,  the  quotient  must 
be  positive  ;  if  one  is  positive  and  the  other  negative,  the  quo- 
tient must  be  negative.     (Art.  99.) 

This  is  manifest  from  the  consideration  that  the  product  of 
the  divisor  and  quotient  must  be  the  same  as  the  dividend. 


If  +aX+b=+ab*) 

C+ab++b=+a 

—aX+b=  —  ab 

—ab-ir-\-b=—a 

7            7   ?   men   < 
+aX—b=—ab  f 

—  ab-. —  &=+a 

—aX—b=+ab) 

+ab~. — b=  —  a 

Div. 

abx      8aXl0ay  —3a3xX6a2      6amXdh 

By 

— a       —2a               —3a2               —2a 

Quot.  —bx     —40ay  —3mXdh=—3dhm 

5.  Divide  2a3bx5yA    by  —a3bxy3. 

6.  Divide  —  33a2bn3x5    by  —  3an3x3. 

118.  If  the  letters  of  the  divisor  are  not  to  be  found  in 
the  dividend,  the  division  is  expressed  by  writing  the  divisor 
under  the  dividend,  in  the  form  of  a  vulgar  fraction. 

_.  XV  *  i       5a 

Thus  xy-r-a= — ;    and  5a-. — h=—r. 


52  DIVISION. 

This  is  a  method  of  denoting  division,  rather  thaa  an  ac- 
tual performing  of  the  operation.  But  the  purposes  of  divis- 
ion may  frequently  be  answered,  by  these  fractional  expres- 
sions. As  they  are  of  the  same  nature  with  other  vulgar 
fractions,  they  may  be  added,  subtracted,  multiplied,  &c. 
See  the  next  section. 

Division  of  Polynomials. 

119.  When  a  simple  factor  is  multiplied  into  a  compound 
one,  the  former  enters  into  every  term  of  the  latter.  (Art.  95.) 
Thus  a  into  b+d,  is  ab+ad.  Such  a  product  is  easily  re- 
solved again  into  its  original  factors. 

Thus  ab+ad=aX(b+d). 

ab+ac-\-ah=aX  (b+c+h). 
amh+amx-\-amy=amX  (h-\-x+y). 
4ad+8ah—12am+4ay=4aX(d+2h—3m+y). 

Now  if  the  whole  quantity  be  divided  by  one  of  these  factors, 
the  quotient  will  be  the  other  factor.     See  Art.  114. 
Thus,  (ab+ad)-r-a=b+d.     Hence, 

If  the  divisor  is  contained  in  every  term  of  a  compound 
dividend,  it  must  be  cancelled  in  each. 

Div.     ab+ac      ahn—ahx      a2h—ay     h2y2—ahy3+hn2y* 
By       a  ah  a  hy2 


Quot.  b-\-c  ah—y 

And  if  there  are  co-efficients,  these  must  be  divided,  in  each 
term  also. 

Div.     Gab—  12a2  c  12d2x—15a2d3y     12hx+S    15xy2—5x* 
By       Sa  3d2  —4  5x 


Quot.  2b  —4ac  shx—2 

ISO.  When  the  divisor  is  not  contained  in  all  the  terms 
of  the  dividend,  the  division  may  be  expressed,  either  by 
placing  the  divisor  under  the  whole  dividend,  or  by  repeating 
it  under  each  term,  taken  separately.     There  are  occasions 


division.  53 

when  it  will  be  convenient  to  exchange  one  of  these  forms 
of  expression  for  the  other. 

b~\~c  b    c 

Thus    b+c  divided  by  x,  is  either  ,  or  -H — . 

And  a+b  divided  by  2,  is  either  — -,  that  is,  half  the  sum 

of  a  and  b;  or  ^+^  that  is,  the  sum  of  half  a  and  half  b. 

For  it  is  evident  that  half  the  sum  of  two  or  more  quantities, 
is  equal  to  the  sum  of  their  halves.  And  the  same  principle 
is  applicable  to  a  third,  fourth,  fifth,  or  any  other  portion  of 
the  dividend. 

So  also  a—b  divided  by  2,  is  either  — -— ,  or  jrwX- 

J  2  2     2 

For  half  the  difference  of  two  quantities  is  equal  to  the 
difference  of  their  halves. 

^     a—2b+h    a     2b     h  .  3a— c        Sa  c 

So  = 1 .      And  = . 

m  m%    m     m  —x        —x      —x 

121.  If  some  of  the  letters  in  the  divisor  are  in  each 
term  of  the  dividend,  the  fractional  expression  may  be  ren- 
dered more  simple,  by  rejecting  equal  factors  from  the  nu- 
merator and  denominator. 


Div. 
By 

ab              n2xy 
ac              hn 

ahm—Say 
ab 

hx+h2n 
hy 

2am 
2xy 

Quot. 

ab         b 
—  or  — 
ac         c 

hm—Sy 
b 

am 
xy 

These  reductions  are  made  upon  the  principle,  that  a  given 
divisor  is  contained  in  a  given  dividend,  just  as  many  times, 
as  double  the  divisor  in  double  the  dividend  ;  triple  the  divisor 
in  triple  the  dividend ;  &c.     See  the  reduction  of  fractions. 

122.  If  the  divisor  is  in  some  of  the  terms  of  the  divi- 
dend, but  not  in  all ;  those  which  contain  the  divisor  may  be 
divided  as  in  Art.  114,  and  the  others  set  down  in  the  form 
of  a  fraction. 

5* 


54  DIVISION. 

™        ,  ,      ^        .     .  ,       ab+d        ab    d  d 

Thus  (ab+ d)  —  a  is  either  ,  or  — h- ,  or  &+-. 

Div.       dxy+rx—hd    Sx2y+nx+b        bm+3y       5a2b—ax 
By         x  x  — b  5a 

Quot.     ay  +r ~OT+3fc 

123.  The  quotient  of  any  quantity  divided  by  itself  or 
its  equal,  is  obviously  a  unit. 

Thus  -=1.     And  - —  =  1.  And— -r=l.  And — -r— -rr=l. 
a  3aa;  4+2  a+6-3A 

Div.    ax+x       6ax2+2x     4axy—4a+8a2d    8a2y— 4x2  —  4 

By      a;  —2a:  4a  4 

Quot.  a+1  #y—  1   +2ad 

Cor.  If  the  dividend  is  greater  than  the  divisor,  the  quo- 
tient must  be  greater  than  a  unit:  But  if  the  dividend  is  less 
than  the  divisor,  the  quotient  must  be  less  than  a  unit. 

124.  A  general  rule  for  division  by  compound  divisors  is 
given  in  Art.  126.  One  case,  however,  deserves  particular 
notice. 

From  what  is  stated  at  the  beginning  of  Art.  119,  it  is- 
obvious,  that 

If  a  compound  expression  containing  any  factor  in  every 
term,  be  divided  by  the  other  quantities  connected  by  their 
signs,  the  quotient  will  be  that  factor. 

Div.    ab+ac+ah  ax2y+hx2y-\-mx2y  6a2y-2axy  2ab-4ax 
By        b+  c+  h  a+h+m  Sa—x  b-2x 

Quot.   a  2ay 

125.  In  division  as  well  as  in  multiplication,  the  caution 
must  be  observed,  not  to  confound  terms  with  factors.  See 
Art.  96. 

Thus     abXac-7-a—a2bc-i-a=abc. 
But     (ab+ac)  +  a=b+c.     (Art.  119.) 
And     abXac~-bXc=a2bc+bc=a2 . 
But     (ab+ac)+(b+c)=a.     (Art.  124.) 


DIVISION.  55 

It  is  a  common  mistake  of  beginners,  to  suppose  that  the 
quotient  of  ab+ac  divided  by  b+c,  is  a+a  instead  of  a. 

136.  When  the  divisor  and  dividend  are  both  polyno- 
mials, the  general  rule  for  performing  the  division  is  the 
following;  which  is  substantially  the  same,  as  the  rule  for 
division  in  arithmetic : 

To  obtain  the  first  term  of  the  quotient,  divide  the  first 
term  of  the  dividend,  by  the  first  term  of  the  divisor  ;# 

Multiply  the  whole  divisor,  by  the  term  placed  in  the  quo- 
tient ;  subtract  the  product  from  a  part  of  the  dividend;  and 
to  the  remainder  bring  down  as  many  of  the  following  terms, 
as  shall  be  necessary  to  continue  the  operation : 

Divide  again  by  the  first  term  of  the  divisor,  and  proceed 
as  before,  till  all  the  terms  of  the  dividend  are  brought  down. 

Ex,  1.     Divide  ac+bc+ad+bd,   by   a+b. 
the  first  subtrahend 
the  second  subtrahend 


Here  ac,  the  first  term  of  the  dividend,  is  divided  by  a,  the 
first  term  of  the  divisor,  (Art.  114.)  which  gives  c  for  the 
first  term  of  the  quotient.  Multiplying  the  whole  divisor  by 
this,  we  have  ac+bc  to  be  subtracted  from  the  two  first  terms 
of  the  dividend.  The  two  remaining  terms  are  then  brought 
down,  and  the  first  of  them  is  divided  by  the  first  term  of 
the  divisor  as  before.  This  gives  d  for  the  second  term  Oi 
the  quotient.  Then  multiplying  the  divisor  by  d,  we  have 
ad+bd  to  be  subtracted,  which  exhausts  the  whole  dividend, 
without  leaving  any  remainder. 

The  rule  is  founded  on  this  principle,  that  the  product  of 
the  divisor  into  the  several  parts  of  the  quotient,  is  equal  to 
the  dividend.  (Art.  113.)  Now  by  the  operation,  the  pro- 
duct of  the  divisor  into  the  first  term  of  the  quotient  is  sub- 

*  See  Note  C. 


ac+bc+ad+bd 
ac+bc 

a+b 
c+d 

the  divisor, 
the  quotient 

*      *     ad+bd 
ad+bd 

56  DIVISION. 

tracted  from  the  dividend ;  then  the  product  of  the  divisor 
into  the  second  term  of  the  quotient ;  and  so  on,  till  the  pro- 
duct of  the  divisor  into  each  term  of  the  quotient,  that  is, 
the  product  of  the  divisor  into  the  whole  quotient,  (Art.  97.) 
is  taken  from  the  dividend.  If  there  is  no  remainder,  it  is 
evident  that  this  product  is  equal  to  the  dividend.  If  there 
is  a  remainder,  the  product  of  the  divisor  and  quotient  is  equal 
to  the  whole  of  the  dividend  except  the  remainder.  And  this 
remainder  is  not  included  in  the  parts  subtracted  from  the 
dividend,  by  operating  according  to  the  rule. 

The  divisor  is  sometimes  set  at  the  left  of  the  dividend,  but 
it  is  more  convenient  to  place  it  on  the  right. 

1  .87.  Before  beginning  to  divide,  it  will  generally  be  ex- 
pedient to  make  some  preparation  in  the  arrangement  of  the 
terms. 

The  letter  which  is  in  the  first  term  of  the  divisor,  should 
be  in  the  first  term  of  the  dividend  also.  And  the  powers  of 
this  letter  should  be  arranged  in  order,  both  in  the  divisor 
and  in  the  dividend;  the  highest  power  standing  first,  the 
next  highest  next,  and  so  on. 

Ex.2.  Divide  2a2b+b3+2ab2+a3,    by    a2+b2+ab. 

Here,  if  we  take  a2  for  the  first  term  of  the  divisor,  the 
other  terms  should  be  arranged  according  to  the  powers  of  a, 
thus, 


a*+2a2b+2ab2+b3 
a3  +  a2b+  ab2 


a2+ab+b2 


a+b 


a2b+  ab2+b* 
a2b+  ab2+b* 


In  these  operations,  particular  care  will  be  necessary  in 
the  management  of  negative  quantities.  Constant  attention 
must  be  paid  to  the  rules  for  the  signs  in  subtraction,  multi- 
plication and  division.     (Arts.  75,  99,  117.) 

Ex.S.  Divide  2ax-2a2x-3a2xy+ Ga3x+axy—xyt  by  2a-y. 


DIVISION.  51 


If  the  terms  be  arranged  according  to  the  powers  of  ay 
they  will  stand  thus ; 


6a*x— 3a2xy— 2a2x+axy+2ax— xy 
6a*x—3a2xy 


2a-y 


3a2x— ax+x 


*     —2a2x+axy 
—2a2x-\-axy 


*        *     +2ax— xy 
+2ax—xy 

Ex.4.  Divide  7xy2+ax2+xA  —  7y2  —  ax2—  x2,  by  x—  1. 

Quot.  x3+ax2+7y2. 
Ex.  5.  Divide  x+6a2h+2a2x— 4a2n+3h— 2n, 

hy3h—2n+x.  Quot.  2a3 +  1. 

Ex.  6.  Divide  7xy3+5x3y—6y*+3x*  —  9x2y2, 

by  2xy+x2  —  3y2.       Quot.  3x2—xy+2y2. 

128.  In  multiplication,  some  of  the  terms,  by  balancing 
each  other,  may  be  lost  in  the  product.  (Art.  104.)  These 
may  re-appear  in  division,  so  as  to  present  terms,  in  the 
course  of  the  process,  different  from  any  which  are  in  the 
dividend. 


a3+x3 

Ex.7. 
a-\-x 

a3+a2x 

a2— ax+x2 

—  a2x 
—a2x 

+x3 
—ax2 

# 

ax2Jrx* 
ax2+x9 

58 


a*—2a*x+2a2x 


DIVISION. 

Ex.  8. 

a2  —  2ax+2x2 


a2  +2ax+2x2 


*  +2a3x— 2a2x2+4x* 
+2a3x—4a2x2  +4ax3 

*      +2a2x2—4ax3+4xA 
+2a2x2—4ax3+4x* 

If  the  learner  will  take  the  trouble  to  multiply  the  quotient 
into  the  divisor,  in  the  two  last  examples,  he  will  find  in  the 
partial  products,  the  several  terms  which  appear  in  the  pro- 
cess of  dividing.  But  most  of  them,  by  balancing  each  other, 
are  lost  in  the  general  product. 

129*  When  there  is  a  remainder  after  all  the  terms  of 
the  dividend  have  been  brought  down,  this  may  be  placed 
over  the  divisor  and  added  to  the  quotient,  as  in  arithmetic. 

Ex.  9. 


ac+bc+ad+bd+x 

a+b 

ac+bc 

a+b 

*     ad+bd 
ad+bd 


Ex.  10. 


ad—ah+bd—bh+y 
ad— ah 


d-h 


a+b+ 


d-h 


*     bd-bh 
bd-bh 


DIVISION. 


59 


It  is  evident  that  a+b  is  the  quotient  belonging  to  the 
whole  of  the  dividend,  excepting  the  remainder  y.  (Art.  126.) 

y 
And   -T^-T   is  tne  quotient  belonging  to  this  remainder. 

(Art.  118.) 

Ex.  11.  Divide  3ab—Gax—bh+2hx+2b2—4bx+c, 
by  b—2x. 


Quot.    3a-/i+2&+ 


6— 2x 


J&z.  12.  Divide  Ga2+4a2-3ax+3a-2x+7,  by  3a+2. 

Quot.  2a2-x+l+^-. 


1.  Div 

2.  Div 

3.  Div 

4.  Div 

5.  Div 

6.  Div 

7.  Div 

8.  Div 

9.  Div 

10.  Div 

11.  Div 

12.  Div 

13.  Div 

14.  Div 

15.  Div 


Promiscuous  Examples. 

de  5x2y—  10ax3  +  15bxy2  —  5x,  by  5x. 

de  12h—  15a+3y2  —3-9bcy+2x,  by  3. 

de  (2+ ax)  (3  —  c)y,  by   (2+ ax)  (3  —  c). 

de  x2y— 3xy— x+2xy2,  by  xy—3y—\-\-2y2. 

de  b2h+nx—by+2a2h—3—b,  by  —5. 

de  ins— 2n2—  aim— 4n-\-x,  by  —  n2x. 

de  2&2  —  fa;2*/— 36x+a:— 9,  by  3bx2y. 

de  4a3#2— a2+6— 2a2a:— 2,  by  —  2a3. 

de  b+x+bnx+nx2,  by  &+#.         Quot.  l+7iar. 

de  l-3a;-f3;r2— a;3,  by  1—  x. 

de  &3+3&2z+3fa;2+:c3,  by  b+x. 

de  3z3  +  16a;2— 5.r+ll,  by  .r+5. 

de  I  — a5,  by  1  — a. 

de  x4—  6x2y2+5y*,  by  x—3y. 

de  6a4-10a3-r-4a-15,  by  3a2-2«+l. 


60  DIVISION. 

130.  A  regular  series  of  quotients  is  obtained,  by  divid- 
ing the  difference  of  the  powers  of  two  quantities,  by  the 
difference  of  the  quantities.     Thus, 

(y2-a2)-T-(y-a)=y+a, 
(y*-a*)  +  (y-a)=y2+ay-\-a2, 
(y*—ak)  +  (y—a)=y2+ay2-\-a2y+a*i 
(y5  — a5)-i-(y— «)=?/4  -\-ay2  +a2y2  +a3y+aA , 
&c. 

Here  it  will  be  seen,  that  the  index  of  y,  in  the  first  term 
of  the  quotient,  is  less  by  1,  than  in  the  dividend;  and  that 
it  decreases  by  1,  from  the  first  term  to  the  last  but  one : 

While  the  index  of  a,  increases  by  1,  from  the  second  term 
to  the  last,  where  it  is  less  by  1,  than  in  the  dividend. 

This  may  be  expressed  in  a  general  formula,  thus, 

(yn--am)-T-(y--a)—ym-l+aym-2 +am-2y+am~1. 

To  demonstrate  this,  we  have  only  to  multiply  the  quotient 
into  the  divisor.     (Art.  113.) 

All  the  terms  except  two,  in  the  partial  products,  will  be 
balanced  by  each  other ;  and  will  leave  the  general  product 
the  same  as  the  dividend. 

Mult,  yt+ayt+atyz+aty+a* 
Into     y  —  a 


ys+ay*+a2y3+a3y2+aAy 
—ayA—a2y3—a:iy2—a*y—a5 

Prod,  y5      *  *  #  *  -a1. 

Mult.  yn-l+ayn-2+a2ym-* +am'ay+am'1 

Into    y— a 


ym-\-aym'l+a2ym'2 +am'2y2+an'"ly 

^-aynt"1—a2ym^2. . .  .—am~*y2—am~1y—a" 


Prod.  ym 


*     —  (. 


DIVISION. 


61 


131.  In  the  same  manner  it  may  be  proved,  that  the  dif- 
ference of  the  powers  of  two  quantities,  if  the  index  is  an 
even  number,  is  divisible  by  the  sum  of  the  quantities.  That 
is,  as  the  double  of  every  number  is  even ; 


(y*™—a*m)^-(yJrd)=y2 


-ay2 


.  .+a27n~2y— a2m~l. 


And  the  sum  of  the  powers  of  two  quantities,  if  the  index 
is  an  odd  number,  is  divisible  by  the  sum  of  the  quantities. 
That  is,  as  2m +1  is  an  odd  number; 


(2/2m+1+«2m+1)^(2/+fl)=2/2m-a2/2 


-a2m'ly+a2nt. 


For  in  each  of  these  cases  the  product  of  the  quotient  and 
divisor,  is  equal  to  the  dividend. 

Thus, 
(y2-a2)-r-(y+a)=y-af 
(y*—aA)~-(y-\-a)=y3—ay2+a2y—a*) 
(y6—a6)~(y+a)=y5—ayA-\-a2y3-—a3y2+aiy—a5, 
&c. 

And* 

(y3+a3)-r-(y+a)=y2-ay+a2, 

(y5+a5)  +  (y+a)=yA--ay3+a2y2—a3y+a*, 

(y'r+a'I)-7-(y+a)=y0-~ay5+a2y*—a3y3+aAy2--a5y+a99 
&c. 


Division  by  Detached  CTo-efficients. 


132.  Division  as  well  as  Multiplication,  may  sometimes 
be  conveniently  performed  by  means  of  detached  co-efficients. 

Suppose  that  2aA+lla3x+20a2x2  +  13ax3+2x*  is  to  be 
divided  by  a2+Sax+2x2. 

6 


62  DIVISION. 

To  perform  the  division  in  the  usual  way,  we  proceed  as 
follows, 


2a*+lla3x+20a2x2+l3ax3+2xA 
2a*  +  6a3x+  4a2  x2 


a2+3ax+2x2 


2a2+5ax  +  x2,  Quot. 


5a3x+16a2x2  +  13ax3 
5a3x+15a2x2  +  10ax3 

a2x2  +  3ax3+2x* 
a2x2+  3ax3-\-2xA 


But  here,  as  the  order  in  which  the  letters  and  exponents 
occur  is  obvious,  we  may  omit  them,  and  proceed  with  the 
detached  co-efficients,  thus ; 


Co-eff.ofdiv'd.     2+11+20+13+2 
2+  6+  4 


1+3+2    co-efF.  of  div'r. 


2+5+1    co-efF.  of  quot 


5+16  +  13 

2a2+5ax-\-x2,  Quot. 
5+15+10 


1+  3+2 
1+  3+2 


After  obtaining  the  co-efficients  of  the  quotient  by  divid- 
ing as  above,  the  proper  letters  and  exponents  are  readily 
supplied.  As  the  first  term  of  the  divisor  is  a2,  and  the  first 
term  of  the  dividend  2a*,  the  first  term  of  the  quotient  must 
be  2a2  ;  and  hence  it  is  evident  that  the  letters  and  expo- 
nents in  the  following  terms  must  be  ax  and  x2 . 

For  the  co-efficients  of  terms  that  are  wanting  in  the  divi- 
sor or  dividend,  ciphers  must  be  substituted.     For  example, 

a5  —  3a3b2+3a2b*+2ab*  —  6b5,  in  which  there  is  no  term 
containing  a*b,  is  divided  by  a3—  ab2+3b3,  in  which  there 
is  no  term  containing  a2  b,  as  follows ; 


1+0-3+3+2-6 
1+0-1+3 


DIVISION.  G3 

1+0-1+3 


1  +  0-2      co-eff.  of  quot. 


°Zl++lXtl  «-«-.  *«*■* 


It  will  be  seen  that  the  second  co-efficient  of  the  quotient 
must  be  0,  because  the  first  remainder  begins  with  0,  under 
the  second  term  of  the  dividend.  And  in  passing  to  the 
third  co-efficient,  we  must  annex  —6,  as  well  as  +2,  to  the 
first  remainder. 

Examples. 

1.  Divide  a4—  a2x2+2ax3  —  #4,   by  a2—  ax+x2. 

2.  Divide  x*—  4x2+6x2—  4z  +  l,  by  z—  L 

3.  Divide  x5+x*y— 5x3y2+6zyA+2y5,  by  x2+3xy+y2. 

4.  Divide  l-4&+10&2-16&3  +  17&4-12&5,  by  1-2&+3&3. 

133.  From  the  nature  of  division  it  is  evident,  that  the 
value  of  the  quotient  depends  both  on  the  divisor  and  the 
dividend;  With  a  given  divisor,  the  greater  the  dividend, 
the  greater  the  quotient.  And  with  a  given  dividend,  the 
greater  the  divisor,  the  less  the  quotient.  In  several  of  the 
succeeding  parts  of  algebra,  particularly  the  subjects  of  frac- 
tions, ratios,  and  proportion,  it  will  be  important  to  be  able  to 
determine  what  change  will  be  produced  in  the  quotient,  by 
increasing  or  diminishing  either  the  divisor  or  the  dividend. 

If  the  given  dividend  be  24,  and  the  divisor  6 ;  the  quotient 
will  be  4.  But  this  same  dividend  may  be  supposed  to  be 
multiplied  or  divided  by  some  other  number,  before  it  is  divi- 
ded by  6.  Or  the  divisor  may  be  multiplied  or  divided  by 
some  other  number,  before  it  is  used  in  dividing  24.  In  each 
of  these  cases,  the  quotient  will  be  altered. 

1*1 1.  In  the  first  place,  if  the  given  divisor  is  contained 
in  the  given  dividend  a  certain  number  of  times,  it  is  obvious 
that  the  same  divisor  is  contained, 

In  double  that  dividend,  twice  as  many  times ; 

In  triple  the  dividend,  thrice  as  many  times ;  &c. 


64  DIVISION. 

That  is,  if  the  divisor  remains  the  same,  multiplying  the 
dividend  by  any  quantity,  is,  in  effect,  multiplying  the  quo- 
tient by  that  quantity. 

Thus,  if  the  constant  divisor  is  6,  then  24 -=-  6=4  the 
quotient. 

Multiplying  the  dividend  by  2,  2X24—6=2X4 

Multiplying  by  any  number  n,  nX24-f-6=7iX4 

135.  Secondly,  if  the  given  divisor  is  contained  in  the 
given  dividend  a  certain  number  of  times,  the  same  divisor 
is  contained, 

In  half  that  dividend,  half  as  many  times ; 

In  one  third  of  the  dividend,  one  third  as  many  times;  &c. 

That  is,  if  the  divisor  remains  the  same,  dividing  the  divi- 
dend by  any  quantity,  is,  in  effect,  dividing  the  quotient  by 
that  quantity. 

Thus,  24-6=  4 

Dividing  the  dividend  by  2,  £24-f-6=j4 

Dividing  by  n,  £24-r-6=£4 

136.  Thirdly,  if  the  given  divisor  is  contained  in  the 
given  dividend  a  certain  number  of  times,  then,  in  the  same 
dividend, 

Twice  that  divisor  is  contained  only  half  as  many  times ; 
Three  times  the  divisor  is  contained  one  third  as  many  times. 

That  is,  if  the  dividend  remains  the  same,  multiplying  the 
divisor  by  any  quantity,  is,  in  effect,  dividing  the  quotient  by 
that  quantity. 

Thus  24-f-6=4 

Multiplying  the  divisor  by  2,  24-r2X6=| 

Multiplying  by  n,  24  -r-  n  X  6  =  ± 


DIVISION.  65 

137*  Lastly,  if  the  given  divisor  is  contained  in  the 
given  dividend  a  certain  number  of  times,  then,  in  the  same 
dividend, 

Half  that  divisor  is  contained  twice  as  many  times ; 
One  third  of  the  divisor  is  contained  thrice  as  many  times 

That  is,  if  the  dividend  remains  the  same,  dividing  the 
divisor  by  any  quantity  is,  in  effect,  multiplying  the  quotient 
by  that  quantity. 

Thus  24-7-  6=4 

Dividing  the  divisor  by  2,  24-7-^6=2x4 

Dividing  by  n,  24-r£6=7iX4 

Resolving  Polynomials  into  Factors. 

138.  Polynomials  may  often  be  rendered  more  conven- 
ient for  certain  purposes,  by  resolving  them  into  factors. 

It  is,  in  many  cases,  easy  to  discover  the  factors  by 
inspection. 

Examples. 

1.  Resolve  h*+3h2x—3hy2  into  factors. 

Ans.  h(h2+3hx-3y2).     See  Art.  119. 

2.  Resolve  4a2— dx2  into  factors. 

Ans.  (2a+3x)  (2a— 3x).     See  Art.  111. 

3.  Resolve  x2+6xy+9y2  into  factors. 

Ans.  (x+Sy)  (x+Sy).     See  Art.  109. 

4.  Resolve  25a2  —  I0ab+b2  into  factors. 

Ans.  (5a -6)  (5a- 6).     See  Art.  110. 
6* 


66  DIVISION. 

5.  Resolve  4x2  +  l2x2y+9xy2  into  three  factors. 

6.  Resolve  x*+y3  into  factors. 

7.  Resolve  1— a3  into  factors. 

8.  Resolve  aA—bA  into  its  factors. 

9.  Resolve  y*  +  l  into  factors. 

10.  Resolve  4x3—4x2-\-x  into  its  factors. 

11.  Resolve  abh2—2abh+ab  into  its  factors. 

12.  Resolve  63nx2y2  +84:nxy-\-28n  into  its  factors. 

13.  Resolve  9h—h  into  three  factors. 

14.  Resolve  3 8  —  1  into  four  factors. 

15.  Resolve  25h2nA—- |-  into  factors. 


FBACTIONS.  67 


SECTION    VI. 


FRACTIONS. 


Art.  139.  Expressions  in  the  form  of  fractions  occur 
more  frequently  in  algebra  than  in  arithmetic.  Most  instan- 
ces in  division  belong  to  this  class.  Indeed  the  numerator 
of  every  fraction  may  be  considered  as  a  dividend,  of  which 
the  denominator  is  a  divisor. 

According  to  the  common  definition  in  arithmetic,  the  de- 
nominator shows  into  what  parts  an  integral  unit  is  supposed 
to  be  divided ;  and  the  numerator  shows  how  many  of  these 
parts  belong  to  the  fraction.  But  it  makes  no  difference, 
whether  the  whole  of  the  numerator  is  divided  by  the  denom- 
inator ;  or  only  one  of  the  integral  units  is  divided,  and  then 
the  quotient  taken  as  many  times  as  the  number  of  units  in 
the  numerator.  Thus  f  is  the  same  as  ^ +J-f  J.  A  fourth 
part  of  three  dollars,  is  equal  to  three  fourths  of  one  dollar. 

140.  The  value  of  a  fraction,  is  the  quotient  of  the  nu- 
merator divided  by  the  denominator. 

Thus  the  value  of  -  is  3.     The  value  of  ■**  is  a. 
Z  o 

From  this  it  is  evident,  that  whatever  changes  are  made 
in  the  terms  of  a  fraction ;  if  the  quotient  is  not  altered,  the 
value  remains  the  same.  For  any  fraction,  therefore,  we 
may  substitute  any  other  fraction  which  will  give  the  same 
quotient. 

mi        4      10      4ba      8drx      6+2     -  ..'v.      ,  . 

ThuS  2  =  T  =  Wa  =  TdVx  =  3+P  &C*  F°r  the  qU°tient 
in  each  of  these  instances  is  2. 

By  the  terms  of  a  fraction  are  meant  the  numerator  and 
/^nominator.  This  use  of  the  word  terms  is  not  to  be  con- 
tended with  the  more  common  one,  explained  in  Art.  38. 
Each  term  of  a  fraction  may  be  a  polynomial.     Thus,  one 

r  a+b  . 
term  of  — ^  is  a+b,  and  the  other  is  c+d. 


68  FRACTIONS. 

141.  As  the  value  of  a  fraction  is  the  quotient  of  the 
numerator  divided  by  the  denominator,  it  is  evident  from 
Art.  123,  that  when  the  numerator  is  equal  to  the  denomina- 
tor, the  value  of  the  fraction  is  a  unit;  when  the  numerator 
is  less  than  the  denominator,  the  value  is  less  than  a  unit; 
and  when  the  numerator  is  greater  than  the  denominator,  the 
value  is  greater  than  a  unit 

The  calculations  in  fractions  depend  on  a  few  general  prin- 
ciples, which  will  here  be  stated  in  connexion  with  each  other. 

14£.  If  the  denominator  of  a  fraction  remains  the  same, 

Multiplying  the  numerator  by  any  quantity,  is  multiplying 
the  value  by  that  quantity ;  and  dividing  the  numerator,  is 
dividing  the  value. 

For  the  numerator  and  denominator  are  a  dividend  and 
divisor,  of  which  the  value  of  the  fraction  is  the  quotient. 
And  by  Art.  134  and  135,  multiplying  the  dividend  is  in  effect 
multiplying  the  quotient,  and  dividing  the  dividend  is  dividing 
the  quotient. 

mi        .       .      n  ab    Sab    7abd   \ab    m 

Ihus  m  the  fractions    — ,    — ,  ,  - — ,  &c. 

a       a        a        a 

The  quotients  or  values  are  b,      3b,     Ibd,    \b,    &c. 

Here  it  will  be  seen  that,  while  the  denominator  is  not 
altered,  the  value  of  the  fraction  is  multiplied  or  divided  by 
the  same  quantity  as  the  numerator. 

Cor.  With  a  given  denominator,  the  greater  the  numera- 
tor, the  greater  will  be  the  value  of  the  fraction ;  and,  on  the 
other  hand,  the  greater  the  value,  the  greater  the  numerator. 

1 13.  If  the  numerator  remains  the  same, 

Multiplying  the  denominator  by  any  quantity,  is  dividing 
the  value  by  that  quantity  ;  and  dividing  the  denominator,  is 
multiplying  the  value. 

For  multiplying  the  divisor  is  dividing  the  quotient ;  and  di- 
viding the  divisor  is  multiplying  the  quotient.    (Art.  136, 137.) 

_      .      r       .         24ab    24ab    24ab    24ab     c 
In  the  fractions  —r- ,    -—r,    -=r~>    ~~T"~>  &c. 
66        12b        3b  b 

The  values  are      4a,        2a,       Sa,      24a,  &c. 


FRACTIONS.  G9 

Cor.  With  a  given  numerator,  the  greater  the  denomina- 
tor, the  less  will  be  the  value  of  the  fraction ;  and  the  less 
the  value,  the  greater  the  denominator. 

111.  From  the  last  two  articles  it  follows,  that  dividing 
the  numerator  by  any  quantity,  will  have  the  same  effect  on 
the  value  of  the  fraction,  as  multiplying  the  denominator  by 
that  quantity;  and  multiplying  the  numerator  will  have  the 
same  effect,  as  dividing  the  denominator. 

1 1*>.  It  is  also  evident  from  the  preceding  articles,  that 
If  the  numerator  and  denominator  be  both  multiplied,  or 

both  divided,  by  the  same  quantity,  the  value  of  the  fraction 

will  not  be  altered. 

^  bx      abx      Sbx     \bx      \abx  ,  ' 

Thus    -7-  =  — r  =  "7TT  =  ~TT  =  TT'  &c-     *  or  m  each  of 
b        ab        3b       -%b        \ab 

these  instances  the  quotient  is  x. 

146.  An  integral  quantity  may,  without  altering  its 
value,  be  converted  into  a  fraction  having  any  proposed  de- 
nominator, by  multiplying  the  quantity  into  this  denominator, 
and  making  the  product  the  numerator. 

_,  a      ab      ad-\-ah      6adh 

Thus   a=T  =  y  =  -j^-  m  W[,  &c.    For  the  quotient 

of  each  of  these  is  a. 

o      *«i     dx+hx       .     ,        •       2dr2+2dr 

So    d+h= .     And  r+l  = —, . 

x  2dr 

147.  There  is  nothing,  perhaps,  in  the  calculation  of 
algebraic  fractions,  which  occasions  more  perplexity  to  a 
learner,  than  the  positive  and  negative  signs.  The  changes 
in  these  are  so  frequent,  that  it  is  necessary  to  become  famil- 
iar with  the  principles  on  which  they  are  made.  The  use 
of  the  sign  which  is  prefixed  to  the  dividing  line,  is  to  show 
whether  the  value  of  the  whole  fraction  is  to  be  added  to,  or 
subtracted  from,  the  other  quantities  with  which  it  is  con- 
nected. (Art.  32.)  This  sign,  therefore,  has  an  influence  on 
the  several  terms  taken  collectively.  But  in  the  numerator 
and  denominator,  each  sign  affects  only  the  single  term  to 
which  it  is  applied. 


70  FRACTIONS. 

The  value  of  -r-  is  a.  (Art.  140.)  But  this  will  become 
negative,  if  the  sign  —  be  prefixed  to  the  fraction. 

mi  ab  n  ab 

lhus    y+-r=y+a.    But   y — r"=y— a- 

So  that  changing  the  sign  which  is  before  the  whole  frac- 
tion, has  the  effect  of  changing  the  value  from  positive  to 
negative,  or  from  negative  to  positive. 

Next,  suppose  the  sign  or  signs  of  the  numerator  to  be 
changed. 

^  ab  -^       —ab 

By  Art.  117,  -r  =  +«-     But  -y-  =  -a. 

.   ah— -he  _       —ab+bc 

And  — 7 —  =  +#— c.     But  r —  =—a+c. 

b  b 

That  is,  by  changing  all  the  signs  of  the  numerator,  the 
value  of  the  fraction  is  changed  from  positive  to  negative,  or 
the  contrary. 

Again,  suppose  the  sign  of  the  denominator  to  be  changed. 

As  before  -r  =  +a.  But  — r  =  —a. 

b  — o 

We  have  then,  this  general  proposition ; 

If  the  sign  prefixed  to  a  fraction,  or  all  the  signs  of  the 
numerator,  or  all  the  signs  of  the  denominator  be  changed; 
the  value  of  the  fraction  will  be  changed,  from  positive  to 
negative,  or  from  negative  to  positive. 

14:8.  From  this  is  derived  another  important  principle. 
As  each  of  the  changes  mentioned  here  is  from  positive  to 
negative,  or  the  contrary ;  if  any  two  of  them  be  made  at 
the  same  time,  they  will  balance  each  other. 

Thus  by  changing  the  sign  of  the  numerator, 

ab  .  —  ab 

~r  =  +a  becomes 


b~~  l      r™*""     b 
But,  by  changing  both  the  numerator  and  denominator,  it 
becomes  "3T"=+«,  where  the-  positive  value  is  restored. 


FRACTIONS.  71 

By  changing  the  sign  before  the  fraction, 

ab  .  ab 

y-\—r=y+d  becomes  y — j-=y—a. 

But  by  changing  the  sign  of  the  numerator  also,  it  becomes 

y —  where  the  quotient  —a  is  to  be  subtracted  from  y, 

or  which  is  the  same  thing,  (Art.  74.)   +  a  is  to  be  added, 
making  y+a  as  at  first.     Hence, 

If  all  the  signs  both  of  the  numerator  and  denominator,  or 
the  signs  of  one  of  these  with  the  sign  prefixed  to  the  whole 
fraction,  be  changed  at  the  same  time,  the  value  of  the  frac- 
tion will  not  be  altered. 


Thl]<.  6_-6_ 

-6 

-   6  -+3 

lhUS  2~-2~ 

A     A        6           -6 

And    32  =  -T  = 

2 

6 
~~2~~ 

-2-+3, 

Hence  the  quotient  in  division  may  be  set  down  in  different 
__  .      .         a      —c         a      c 

ways.     Thus  (a— c)-=-&,  is  either  t  +  ""t~>  or  t~t- 

The  latter  method  is  the  most  common.  See  the  exam- 
ples in  Art.  122. 

Reduction  of  Fractions. 

1 19.  From  the  principles  which  have  been  stated,  are 
derived  the  rules  for  the  reduction  of  fractions,  which  are 
substantially  the  same  in  algebra,  as  in  arithmetic. 

A  fractioii.  may  be  reduced  to  lower  terms,  by  dividing 
both  the  numei^tor  and  denominator,  by  any  quantity  which 
will  divide  them  without  a  remainder. 

According  to  Art.  145,  this  will  not  alter  the  value  of  the 
fraction. 

___.        ab      a  _  6dm      3m  ■    7m       1 

lhus  -t  =  --     And  ■7rj-  =  T_-     And  - —  =  -. 

co      c  Say      Ay  Imr      r 


72  FRACTIONS. 

In  the  last  example,  both  parts  of  the  fraction  are  divided 
by  the  numerator. 

a+bc         1        .   \  am+ay     a 

•  Again,  - — —r\—  =~~ •     And  7 — rT~==T# 

(a-\-bc)m     m  bm+by     b 

If  a  letter  is  in  every  term,  both  of  the  numerator  and  de- 
nominator, it  may  be  cancelled;  for  this  is  dividing  by  that 
letter.     (Art.  119.) 

Sam+ay     Sm+y      dry-\-dy      r-f-1 


Thus, 


ad+ah     '    d+h'     dhy—dy      /i— 1 


If  the  numerator  and  denominator  be  divided  by  the  great- 
est common  measure,  it  is  evident  that  the  fraction  will  be 
reduced  to  the  lowest  terms.  For  the  method  of  finding  the 
greatest  common  measure,  see  Sect.  XV. 

150.  Fractions  of  different  denominators  may  be  reduced 
to  a  common  denominator,  by  multiplying  each  numerator 
into  all  the  denominators  except  its  own,  for  a  new  numerator  ; 
and  all  the  denominators  together,  for  a  common  denominator. 

__  _    .        a       _  c  m 

Ex.  1.  Reduce  t>  and  -3.  and  — ,  to  a  common  denominator. 
V         d  y 

axdxy=ady\ 

cXbXy=cby  >  the  three  numerators. 

mXbXd—mbd) 

bxdXy—bdy       the  common  denominator. 

mi      r  1  ady         ,    bey         .  bdm 

The  fractions  reduced  are  s-7-1  and  7-7-,  and  7-7-. 

bdy  bay  bdy 

Here  it  will  be  seen,  that  the  reduction  consists  in  multi- 
plying the  numerator  and  denominator  of  each  fraction,  into 
all  the  other  denominators.  This  does  not  alter  the  value. 
(Art.  145.) 

"    _    ,  ab         ,371s  g 

2.  Keduce    —r,  and  ,  and  — . 

2h  m  Sx 

3.  Reduce    -,  and  — ,  and  — — . 

4  y2  x~\-2 

2  ° 

4.  Reduce    — — »  and 


x+2  x—2 


FRACTIONS.  73 

After  the  fractions  have  been  reduced  to  a  common  de- 
nominator, they  may  be  brought  to  lower  terms,  by  the  rule 
in  the  last  article,  if  there  is  any  quantity  which  will  divide 
the  denominator,  and  all  the  numerators  without  a  remainder. 

An  integer  and  a  fraction,  are  easily  reduced  to  a  common 
denominator.     (Art.  146.) 

__,  _  b  .        a        ;  b  ac  h 

Thus  a  and  -  are  equal  to  -  and  -,  or  —  and  — 
c  ^  \  c  c  c 

h     d  amy    bmy     hy     dm 

And  a,  b.  —  >  -  are  equal  to  >   >    — ■    — 

my  my      my     my     my 

151.  To  reduce  an  improper  fraction  to  an  integral  or 
a  mixed  quantity,  divide  the  numerator  by  the  denominator ;  for 
the  integral  part ;  and  if  there  is  a  remainder,  place  this  over 
the  denominator  for  the  fractional  part.     See  Art.  122. 

_.        ab+b?n-\-d  d 

lhus  : =a-\-m+T- 

b  b 

_    :         6xy—4x2+2x—3h2  .      , 

Reduce ,  to  a  mixed  quantity. 

<&x 

For  the  reduction  of  a  mixed  quantity  to  an  improper  frac- 
tion, see  Art.  154.  And  for  the  reduction  of  a  compound 
fraction  to  a  simple  one,  see  Art.  163.    . 


Addition  of  Fractions. 

152.  In  adding  fractions,  we  may  either  write  them  one 
after  the  other,  with  their  signs,  as  in  the  addition  of  integers, 
or  we  may  incorporate  them  into  a  single  fraction,  by  the 
following  rule : 

Reduce  the  fractions  to  a  common  denominator,  make  the 
signs  before  them  all  positive,  then  add  their  numerators,  and 
place  the  sum  over  the  common  denominator. 

The  common  denominator  shows  into  what  parts  the  inte- 
gral unit  is  supposed  to  be  divided  ;  and  the  numerators  show 
the  number  of  these  parts  belonging  to  each  of  the  fractions. 
(Art.  139.)  Therefore  the  numerators  taken  together  show 
the  whole  number  of  parts  in  all  the  fractions 

7 


74  FRACTIONS. 

2       1       1        :     .    3       111 
Thus,   -  =  -  +  -.     And  -  =  -  +  -  +  -. 

2      3       111115 

Therefore,    -  +  -==-  +  ^  +  -  +  -  +  ^  =  -- 

The  numerators  are  added,  according  to  the  rules  for  the 
addition  of  integers.     (Art.  60,  &c.) 

To  avoid  the  perplexity  which  might  be  occasioned  by  the 
signs,  it  will  be  expedient  to  make  those  prefixed  to  the  frac- 
tions uniformly  positive,  as  the  rule  directs.  But  in  doing 
this,  care  must  be  taken  not  to  alter  the  value.  This  will  be 
preserved,  if  all  the  signs  in  the  numerator  are  changed  at 
the  same  time  with  that  before  the  fraction.     (Art.  148.) 

Ex.  1.  Add    —  and  —  of  a  pound.     Ans.   -— - •  or  — • 
lb  lb  lb  lb 

It  is  as  evident  that  fa,  and  fa  of  a  pound,  are  fa  of  a 
pound,  as  that  2  ounces  and  4  ounces,  are  6  ounces. 

a  c 

2.  Add  -r  and  -y     First  reduce  them  to  a  common  denomi- 

b         a 

mi  mi    i        ,     ad        .be         _    ■    .  ad+bc 

nator.     lhey  will  then  be  j-j  and  7-3  >  and  their  sum  — j-z — 

/-..  m        i        2r+d  . 

3.  Given   -y  and —r—>  to  find  their  sum. 

a  3ri 

m       _      2r+d    3hm       .       2dr+d2      3hm—2dr—d2 

Ans.  -T  and —j—  =-^rr  and t-jj — = —77 

d  3h        Sdh  3dh  Sdh 

b_  h2-\       b        -A2 +1  __  ab-2h2x+2x 

2x  a      ~~  2x  a  2ax 

r    2       _     3b        -2x2       3ab       -2x2 +3ab      2x2 -3ab 

5.  —  and  = 1 = or -• 

a  —x2      —ax2     —ax2  —ax2  ax2 

y         ,     x       y2—xy+xy+x2     y2+x2 

6.  -7-  and «Z Z — 1 ^ _.     (Art.  70.) 

y+x         y—x  y2—x2  y2—x2       v  ' 

'••-  —  *         —x2 

7.  Add  —  to 

2y         1—n 

.  11  ~12  —6 

8.  Add  — ^—  to  - — -  •  Ans.  —7. 

o  7  —  o 


FRACTIONS.  7«> 

153.  For  many  purposes,  it  is  sufficient  to  add  fractions 
in  the  same  manner  as  integers  are  added,  by  writing  them 
one  after  another  with  their  signs.     (Art.  60.) 

mi         i  r  a         i3        i  d     .     a      3       d 

lhus  the  sum  of  r  and  -  and  —  —  >  is  r  H — 

o  y  2m        o      y     2m 

In  the  same  manner,  fractions  and  integers  may  be  added. 

_n  r  d  _       h    .  d     h 

The  sum  of  a  and  -  and  Sm  and >  is  a+Sm-\ 

y  r  V     r 

15  I.  Or  the  integer  may  be  incorporated  with  the  frac- 
tion, by  converting  the  former  into  a  fraction,  and  then  add- 
ing the  numerators.     See  Art.  146. 

_.  r  _    b     .    a       b      am       b      am+b 

I  he  sum  of  a  and  — >  is  -  H —  = 1 —  = 

m         \      m      m       m         m 

-«,  „  ,     A+rf     .    3my— 6v2+A+^ 

1  he  sum  of  Sy  and  —  ?  is 

J  m—2y  m—2y 

Incorporating  an  integer  with  a  fraction,  is  the  same  as 
reducing  a  mixed  quantity  to  an  improper  fraction.  For  a 
mixed  quantity  is  an  integer  and  a  fraction.  In  arithmetic, 
these  are  generally  placed  together,  without  any  sign  between 
them.  But  in  algebra,  they  are  distinct  terms.  Thus  2\  is 
2  uiid  J,  which  is  the  same  as  $£-]-§-. 

The  rule  for  reducing  a  mixed  number  to  an  improper 
fraction,  may  be  thus  stated : 

Multiply  the  integer  into  the  denominator  of  the  fraction, 
add  the  product  to  the  numerator,  and  place  the  sum  over 
the  denominator. 

If  the  sign  before  the  fraction  is  — ,  it  must,  before  apply- 
ing the  rule,  be  changed  to  + ;  and  in  order  to  leave  the 
value  of  the  fraction  unaltered,  the  signs  of  the  numerator 
or  of  the  denominator  must  also  be  changed.     See  Art.  152 

Ex.  1.  Reduce  2 — •  Ans. 

a  a 

n    t>   j  ,3  a—ab+b—4 

2.  Reduce  a-\- - — =-.  Ans.  ■ — t 

l  —  o  1—b 


7fi  FRACTIONS. 

_    \  7      Sx  .         2by+Sx 

3.  Reduce  2b -\ Ans.  — * 

y  y 

4.  Reduce  n— -• 

9 

2x2 1 

5.  Reduce   1  -\ • 

5x 

Sx 


6.  Reduce  2y 


y-2a 

Subtraction  of  Fractions. 

155.  The  methods  of  performing  subtraction  in  algebra, 
depend  on  the  principle,  that  adding  a  negative  quantity  is 
equivalent  to  subtracting  a  positive  one  ;  and  v.  v.  (Art.  74.) 
For  the  subtraction  of  fractions,  then,  we  have  the  following 
simple  rule. 

Change  the  fraction  to  be  subtracted,  from  positive  to 
negative,  or  the  contrary,  and  then  proceed  as  in  addition. 
(Art  152.) 

In  making  the  required  change,  it  will  be  expedient  to  alter, 
in  some  instances,  the  signs  of  the  numerator,  and  in  others, 
the  sign  before  the  dividing  lin*\  (Art  id.7  )  ™  •?«  tn  leave  the 
latter  always  positive. 

Ex.  1.  From  t>  subtract  — 
b  m 

First  change  —  >  the  fraction  to  be  subtracted,  to 

Secondly,  reduce  the  two  fractions  to  a  common  denomi- 

am  —  bh 

nator,  making,  ^  and  -^-  ■ 

Thirdly,  the  sum  of  the  numerators  am—bh,  placed  over 

am  —  bh 
the  common  denominator,  gives  the  answer,  — j 

-^  1  —  V  2  A        x—xy—Aa 

2.  From  — —    subtract  -•  Ans.  ^r— 

2a  x  2ax 


2n 


FRACTIONS. 

2a- 1 


3.  From  —  subtract 

y  * 

..         2a— 3       _  a— 5 

4.  From  — - —   subtract 


1—x2 

5.  From  — - —  subtract 

2y 

1  x 

6.  From  -  subtract  -• 

a  2 

„         x+2      .  y— 2 

7.  From  subtract 

y 


Ans. 


77 
2na:— 2ay+y 


xy 


Ans. 


19-a 


10 

3+2y-2xy-3x* 


156.  Fractions  may  also  be  subtracted,  hUnClegers,  by 

setting  them  down,  after  their  signs  are  changed,  without  re- 
ducing them  to  a  common  denominator. 


k      _  h+d 

r  rom  —  subtract 

m  y 


Ans. 


h      h+d 


In  the  same  manner,  an  integer  may  be  subtracted  from  a 
fraction,  or  a  fraction  from  an  integer. 


From  a  subtract  —  • 
m 


Ans.  a—  ■ 


m 


157.  Or  the  integer  may  be  incorporated  with  the  frac- 
tion, as  in  Art.  154. 


Ex.  1.  From  -  subtract  2y. 


o 

2.  From  3x— -  subtract  — 2a;+-    Ans. 
2  y 


2            2—2xy 
Ans.  — 2y= 

10xy-ny-6 


-r.         0—2  ,  2— a 

3.  From 1  subtract 


Ans. 


2y 

2a— 4— x 


^                        1— 2a       .  i;    ,  fl-2 

4.  From  2y+# - — >  subtract  y— 2x+  — =— 


78  fractions. 

Multiplication  of  Fractions. 

158.  By  the  definition  of  multiplication,  multiplying  by  a 
fraction  is  taking  a  part  of  the  multiplicand,  as  many  times 
as  there  are  like  parts  of  an  unit  in  the  multiplier.  (Art.  85.) 
Now  the  denominator  of  a  fraction  shows  into  what  parts 
the  integral  unit  is  supposed  to  be  divided ;  and  the  numera- 
tor shows  how  many  of  those  parts  belong  to  the  given  frac- 
tion. In  multiplying  by  a  fraction,  therefore,  the  multipli- 
cand is  to  be  divided  into  such  parts,  as  are  denoted  by  the 
denominator ;  and  then  one  of  these  parts  is  to  be  repeated, 
as  many  times,  as  is  required  by  the  numerator. 

Suppose  a  is  to  be  multiplied  by  -  • 


a 
A  fourth  part  of  a  is  -  - 
4 


mi  •    .  t       «    •         .    a      a      a     3a 

This  taken  3  times  is  7  +  7  +  7  =  -r '     (Art  152) 


4      4      4       4 
a  .        .  , 3 


Again,  suppose  r  is  to  be  multiplied  by    - 

One  fourth  of  r  is  jr-     (Art.  143.) 

mi.'    j.  i       «  .•         .     a   ,   a       a     3a 
This  taken  3  times  is  tt  +  tt  +  -7=:— r> 
4b     4b      4b     4b 

the  product  required. 

In  a  similar  manner,  any  fractional  multiplicand  may  be 
divided  into  parts,  by  multiplying  the  denominator;  and  one 
of  the  parts  may  be  repeated,  by  multiplying  the  numerator. 
We  have  then  the  following  rule  : 

To  multiply  fractions,  multiply  the  numerators  together, 
for  a  new  numerator,  and  the  denominators  together,  for  a 
new  denominator. 

Ex.  1.  Multiply  S  into  ± .  Product  p. 

c  Am  2cm 


FRACTIONS.  79 

~    ™  i  •  i     a+d  4h  -n     j         4ah+4dh 

2.  Multiply  into -•  Product  — . 

r  J      y  m—2  my—2y 

«•  ,  .  ,     h(a+m)  .  4  ^     ,        4h(a-\-m) 

3.  Multiply  -v  0     ;  into  7 c-  Product  -^7 r*- 

r  J         3  (a— n)  3(a—n) 

*«  1  .  1     2— a;  .        «— & 

4.  Multiply  7-7—  into  -. 

rj   1+2/  2/-2 

5.  Multiply  —  into  -• 

1  oO.  The  method  of  multiplying  is  the  same,  when  there 
are  more  than  two  fractions  to  be  multiplied  together. 

.,,,.,  .       a     c         ,77i  _.     ,        acm 

1.  Multiply  together  r>    -v  and  —  Product  -7-7-- 

CL      C  CtC  m         CLC7TI 

For  tXj  is,  by  the  last  article  r-z,  and  this  into  —  is  7-7-- 
b     d         J  bd  y       bdy 

_,_,.,     x    2y—l      7i        3  •'•'._  6nxy— Snx 

2.  Multiply  -,  3L_,    -h,—b.     Product  j^^ 

3.  Multiply  — ,  -  and  -g*. 

TIT     ,    •     1         3*       5  J     !+2& 

4.  Multiply  — f  -  and  j^- 

160*  The  multiplication  may  sometimes  be  shortened, 
by  rejecting  equal  factors,  from  the  numerators  and  denom- 
inators. 

Tt/r  1  .  1     a  .         h       .  d  _     _        <Z/* 

1.  Multiply  -  into  -  and  -•  Product  — 

r  J   r  a  y  ry 

Here  <z,  being  in  one  of  the  numerators,  and  in  one  of  the 
denominators,  may  be  omitted.     If  it  be  retained,  the  pro- 

adh 
duct  will  be  — •      But   this   reduced   to  lower   terms,    by 

ary  J 

Art.  149,  will  become  —  as  before. 
ry 


80  FRACTIONS. 

Tt/r  ,  .  ,      2    .         ab  ay  _     .         2<2& 

2.  Multiply  —  into  —  and  -~  Product  — • 

ax  Sy  O  <3X 

It  is  necessary  that  the  factors  rejected  from  the  numera- 
tors be  exactly  equal  to  those  which  are  rejected  from  the 
denominators.  In  the  last  example,  a  being  in  two  of  the 
numerators,  and  in  only  one  of  the  denominators,  must  be 
retained  in  one  of  the  numerators. 

„-.-■-     a— b  hx  L     .         ah—bh 

3.  Multiply  into  -f-  Product  — 

xy  ocl  *$@y 

Here,  though  the  same  letter  a  is  in  one  of  the  numera- 
tors, and  in  one  of  the  denominators,  yet  as  it  is  not  in  every 
term  of  the  numerator,  it  must  not  be  cancelled. 

,,  ,  .  ,     2xy—  1  .         n  2a 

4.  Multiply  — into  —  and  — 

a  Sx  y 

If  any  difficulty  is  found,  in  making  these  contractions,  it 
will  be  better  to  perform  the  multiplication,  without  omitting 
any  of  the  factors ;  and  to  reduce  the  product  to  lower  terms 
afterwards. 

161.  When  a  fraction  and  an  integer  are  multiplied  to- 
gether, the  numerator  of  the  fraction  is  multiplied  into  the 
integer.  The  denominator  is  not  altered ;  except  in  cases 
where  division  of  the  denominator  is  substituted  for  multipli- 
cation of  the  numerator,  according  to  Art.  144. 

mi  v.772    am  ta  a  ,  a    m     am 

Thus  aX-= —        For  a=T;   and  -X— = 

y      y  1  1     y      y 

x     h+1      2hnx+2nx  ,         1      a 

So  2wx_x__  =  ___ And  ax_  =  _. 

The  fraction  maybe  multiplied,  by  dividing  its  denomina- 
tor instead  of  multiplying  the  numerator ;  and  this  method 
is  to  be  preferred,  whenever  the  division  can  be  actually 
performed. 

Thus,  to  multiply  —  by  x}  divide  the  denominator  by  a\ 

The  result  is  — - 
2y 


FRACTIONS.  81 

1 62.  A  fraction  is  multiplied  into  a  quantity  equal  to 
its  denominator,  by  cancelling  the  denominator. 

Thus,  to  multiply  the  fraction  r  by  6,  divide  its  denomin- 
ator by  b.     The  result  is  -  or  a. 

n      i  i  n    3m    .  .    3m 

So  the  product  of  -— -  into  a— y  is  —  or  Sm. 

.  h+3d     ,         .      _ 
And  — - —  X(3+m)=ft+3i. 
3+m     v         y 

On  the  same  principle,  a  fraction  is  multiplied  into  any 
factor  in  its  denominator,  by  cancelling  that  factor. 

_.         a         .a 
Thus  j—Xy  is  y 

For  cancelling  ?/  in  the  denominator  is  dividing  the  denom- 
inator by  y ;  and  dividing  the  denominator  is,  in  effect,  mul- 
tiplying the  fraction. 

So  — X6^7-         And     ,  ,  uhX(a+b)=T 
24  4  ah+bh     v        '     h 

163.  From  the  definition  of  multiplication  by  a  fraction, 
it  follows  that  what  is  commonly  called  a  compound  fraction  * 

is  the  product  of  two  or  more  fractions.     Thus  -ofr  is  tXt* 

_       3     _  <z  .    1     .  a     _        .  .  .       .       a       a       a 

t  or  -  oi  r>  is  T  ol  t  taken  three  times,  that  is,  77  +  77  +  ti- 
4       6       4        6  46     46     46 

But  this  is  the  same  as  t  multiplied  by  -•     (Art.  158.) 

Hence,  reducing  a  compound  fraction  into  a  simple  one,  is 
the  same  as  multiplying  fractions  into  each  other. 

Ex.  1.  Reduce  -  of  r-rs:-  Ans. 


7        6+2  *   76+14 

2.  Reduce    7  of  -  of  -•  Ans.   -- 


4        5        Ti—2  '    20n-40 


*  By  a  compound  fraction  is  meant  a  fraction  of  a  fraction,  and  not  a  frac- 
tion whose  numerator  or  denominator  is  a  compound  quantity. 


82  FRACTIONS. 

3.  Reduce   -  of  -  of  x -*•  Ans. 


3        7        2a— 3b  '  42a-63& 

164.  The  expression  fa,  \b,  %y,  &c.  are  equivalent  to 
2a    b    Ay      _       .      .     0     L  . ;  .   .  .        2  2a 
~o~'  P  V     *or  3a  1S  f  °*  a>  which  is  equal  to  -Xfl=T- 
o      y     7                                                                                     o  «i 

(Art.  161.)     So  }&=ix&=£- 

Division  of  Fractions. 

1 65.  To  divide  one  fraction  by  another,  invert  the  divi- 
sor, and  then  proceed  as  in  multiplication.     (Art.  158.) 

a         c 
Let  it  be  required  to  divide  t  by  -v     If  we  begin  by  di- 
viding by  c  only,  the  quotient,  according  to  Art.  143,  will  be 
t—     Then  if  the  divisor,  instead  of  being  c,  is  c  divided  by  d, 

the  quotient  must,  according  to  Art.  137,  be  d  times  r-,  that 

is,  (Art.  161.)  T-;  and  this  is  the  product  of  j  the  dividend, 

into  -j    the  divisor  inverted.      Hence  we  derive  the  rule 
c 

stated  above. 

The  division  may  be  proved,  by  multiplying  the  divisor 
and  quotient  together.  The  product  should  be  equal  to  the 
dividend.     (Art.  113.) 

__    _.  .,  h2   _  2h  h2      a     ah2     ah 

Ex.  1.  Divide  —  by  —      Ans.  Tr^TTL—'^r—'^' 
3y     J    a  3y    2h     6hy    6y 

__   „  ah    2h      h2 
Proof.  —  X— =— • 
6y     a      3y 

_,  .,  2y-3  ,  a  k        2y-3     by     10*/2-15y 

2.  Divide  ~ by  — •   Ans.  -^ — X— =— -• 

x        J   5y  x         a  ax 

«  n    10ya  — 15y  a     2y-3 

Proof.  — - -X  —  =  — 

ax  5y        x 


TRACTIONS.  83 

m    —   .,     an2          nx  an2     3a     a2n 

3.  Divide  —  by  —  •  Ans.   — -x  — = 

Sy      *   3a  3y      nx      xy 

_.       _   a2n  nx     an2               . 

Proof.  x— =  ^r—  the  dividend. 

xy  3a      3y 

_.  ..     20a3  .      5a  20a3     3# 

4.  Divide  — -  by  — •  Ans.  — —  X-r-=4a2#. 

3         J   3a;  3        5a 

„    _.   . .     2a2— x  .      a2—  2x 

5.  Divide  — ^t —  by  • 

2h         J        n 


n      TV     *J        «6+l     U  2 

6.  Divide  — - —  by  —7 — ;• 
3         J    ab—l 

166.  When  a  fraction  is  divided  by  an  integer,  the  a*e- 
nominator  of  the  fraction  is  multiplied  into  the  integer. 

Thus  the  quotient  of  t  divided  by  m,  is  s— •• 

«  wi         ,1,,  .  .     a    m    a     1       a 

ror  m=— ;  and  by  the  last  article,  1~=tX-=jt- 
1  J  o     1     b     m     bm 

1,11  1  ,3  3       1 

So   7-r-h= tXt  =  -t — 77-         And  7-7-6=— =  -• 

a— &  a— 6     ft     ah—bh  4  24      8 

See  Art.  143. 

And  an  integer  is  divided  by  a  fraction,  by  multiplying  the 
integer  into  the  fraction  inverted. 

Thus  the  quotient  of  a  divided  by  ->  is  •?-• 

a  a      b      a    c      ac 

For   a—-\  and  tt-=1-X7  =  t- 
1  1      c     1     b      b 

In  fractions,  multiplication  is  made  to  perform  the  office 
of  division  ;  because  division  in  the  usual  form  often  leaves  a 
troublesome  remainder :  but  there  is  no  remainder  in  multi- 
plication. 

In  many  cases,  there  are  methods  of  shortening  the  opera- 
tion. But  these  will  be  suggested  by  practice,  without  the 
aid  of  particular  rules. 


84  FRACTIONS. 

167.  By  the  definition,  (Art.  43.)  "the  reciprocal  of  a 
quantity,  is  the  quotient  arising  from  dividing  a  unit  by  that 
quantity." 

a  a  b      b 

Therefore  the  reciprocal  of  r  is  1 .  -r  r  =  1 X  -  =  -•     1  nat  is, 

The  reciprocal  of  a  fraction  is  the  fraction  inverted. 
Thus  the  reciprocal  of  — —  is  -y-  ;    the   reciprocal  of 
—  is  -j-  or  Sy ;  the  reciprocal  of  J  is  4. 

Hence  the  reciprocal  of  a  fraction  whose  numerator  is  1, 
is  the  denominator  of  the  fraction. 

Thus  the  reciprocal  of  -  is  a;  of        ,»  is  a+b,  &c. 

168.  A  fraction  sometimes  occurs  in  the  numerator  or 

2 

denominator  of  another  fraction,  as  -r-»     It  is  often  conven- 
er 

ient,  in  the  course  of  a  calculation,  to  transfer  such  a  frac- 
tion from  the  numerator  to  the  denominator  of  the  principal 
fraction,  or  the  contrary.  That  this  may  be  done  without 
altering  the  value,  if  the  fraction  transferred  be  inverted,  is 
evident  from  the  following  principles : 

First,  Dividing  by  a  fraction,  is  the  same  as  multiplying 
by  the  fraction  inverted.     (Art.  165  and  166.) 

Secondly,  Dividing  the  numerator  of  a  fraction  has  the 
same  effect  on  the  value,  as  multiplying  the  denominator ; 
and  multiplying  the  numerator  has  the  same  effect,  as  divid- 
ing the  denominator.     (Art.  144.) 

— a  a 

Thus  in  the  expression  —  the  numerator  of  -  is  multiplied 

into  f.  But  the  value  will  be  the  same,  if,  instead  of  multi- 
plying the  numerator,  we  divide  the  denominator  by  f,  that 
is,  multiply  the  denominator  by  f . 

mi       r        ia       a  h       Vi 

Therefore  —  ==  — .         So  — -  =  — 

x       \x  \m      m 

a    a     $d             d               d                a    j  am~x      ia—lx 
And  t— —  =  77iri — \=  u  ,  A  •        And  — — as- — • 


FRACTIONS.  85 

160.  Multiplying  the  numerator,  is  in  effect  multiplying 
the  value  of  the  fraction.  (Art.  142.)  On  this  principle,  a 
fraction  may  be  cleared  of  a  fractional  co-efficient  which 
occurs  in  its  numerator. 


_.        fa      3     a      3a 

Thus  T  =  5xl=Tb- 

,  jfl      la      a 
And  ±-  =  -x-  =  —- 
y       5    y      5y 

.  ih-\-\x     1     h+x 

And  * L=-X = 

m          3       m 

h+x                ■   ix       Sx 

=  — And  r-  =  ;rr- 

3m                     5a     20a 

~      ,        ,       .       .    3a      3     a      la 
On  the  other  hand,  — -  =  -X  -  =  - — 
7a:      7     x       a; 

,    a       1     a     \a  4a 

And  —  =  -x-==£-         And 


%y    3   y     y  5d+5x    eH-a: 

170.  But  multiplying  the  denominator \  by  another  frac- 
tion, is  in  effect  dividing  the  value ;  (Art.  143.)  that  is,  it  is 
multiplying  the  value  by  the  fraction  inverted.  The  princi- 
pal fraction  may  therefore  be  cleared  of  a  fractional  co- 
efficient which  occurs  in  its  denominator. 

mt         a      a      3      a    5      5a  _    a       7a 

Thus  pnti^iTsr    And  p^e- 

,   a+A-    9a+9£  .   3A      21* 

And  -; —  = — And  t-~=- — 

%y  2y  \m      4m 


I 


On  the  other  hand, 


7a       a 

3x  """  ##* 


,   3y-f303:      y+d#  ,  3x      x 

And  -^ =  —2 And  —  «  r- 

2w  fm  y      ty 

171.  The  numerator  or  the  denominator  of  a  fraction, 
may  be  itself  a  fraction.  The  expression  may  be  reduced 
to  a  more  simple  form,  on  the  principles  which  have  been 
applied  in  the  preceding  cases. 

a  x 

m.       ~b     a     c      ad  y      x  r      nr 

Thus  -=,—-  =  —.         And  t  =  t—         And  ~-=ZT 
c      b     d      be  h     hy  mm 


S6  SIMPLE      EQUATIONS. 


SECTION    VII. 


SIMPLE    EQUATIONS. 

Art.  1 72.  The  subjects  of  the  preceding  sections  are  in- 
troductory to  what  may  be  considered  the  peculiar  province 
of  algebra,  the  investigation  of  the  values  of  unknown  quan- 
tities by  equations. 

An  equation  is  a  proposition,  expressing  in  algebraic 
characters,  the  equality  between  one  quantity  or  set  of  quan- 
tities and  another,  or  between  different  expressions  for  the 
same  quantity  * 

'  Thus  x-\-a=b+c,  is  an  equation,  in  which  the  sum  of  x 
and  a,  is  equal  to  the  sum  of  b  and  c.  The  quantities  on  the 
two  sides  of  the  sign  of  equality,  are  sometimes  called  the 
members  of  the  equation ;  the  several  terms  on  the  left  con- 
stituting the  first  member,  and  those  on  the  right,  the  second 
member. 

17.1.  An  identical  equation  is  one  in  which  the  two 
members  are  of  the  same  form,  or  are  such  as  may  be  re- 
duced to  the  same  form. 

Sx—ax+2=Sx—ax+2,     (x—1)  (x-2)=x2—3x+2, 

are  equations  of  this  kind. 

An  equation  that  is  identical,  will  hold  good  for  any  values 
that  may  be  substituted  for  the  letters  contained  in  it, 

Thus,  in  the  last  equation,  if  x  equal  10,  each  of  the  mem- 
bers will  be  72  ;  and  if  x  equal  50,  each  member  will  be  2352. 

But  an  equation  which  is  not  identical,  will  not  be  satisfied 
except  for  particular  values  of  the  letters. 

Thus  the  equation  x2=3x—2  will  be  satisfied,  if  we  sub- 
stitute either  1  or  2  for  x ;  but  if  we  substitute  3,  4,  or  any 
other  number,  the  equation  will  be  destroyed. 

1 74t.  An  equation  containing  only  the  first  power  of  an 
unknown  quantity  is  called  a  simple  equation,  or  an  equation 

*  See  Note  D. 


SIMPLE      EQUATIONS.  87 

of  the  first  degree.  One  in  which  the  highest  power  of  the 
unknown  quantity  is  a  square,  is  called  a  quadratic  equation, 
or  an  equation  of  the  second  degree.  And  in  general,  the 
degree  of  an  equation  is  marked  by  the  highest  power  of  the 
unknown  quantity  contained  in  the  equation. 

Thus  ax=c,  ax2=bx-\-c,  ax3+bx2=c,  xx -\-2x3=x-\-a, 
are  equations  of  the  first,  second,  third  and  fourth  degrees ; 
and  are  called  also  simple,  quadratic,  cubic  and  biquadratic 
equations. 

It  will  here  be  convenient  to  attend  only  to  simple  equa- 
tions, containing  one  unknown  quantity. 

1 7o.  The  object  aimed  at,  in  what  is  called  the  resolution 
or  reduction  of  an  equation,  is  to  find  the  value  of  an  un- 
known quantity.  In  the  first  statement  of  the  conditions  of 
a  problem,  the  known  and  unknown  quantities  are  frequently 
thrown  promiscuously  together.  To  find  the  value  of  that 
which  is  required,  it  is  necessary  to  bring  it  to  stand  by  it- 
self, while  all  the  others  are  on  the  opposite  side  of  the  equa- 
tion. But  in  doing  this,  care  must  be  taken  not  to  destroy 
the  equation,  by  rendering  the  two  members  unequal.  Many 
changes  may  be  made  in  the  arrangement  of  the  terms,  with- 
out affecting  the  equality  of  the  sides. 

176.  The  reduction  of  an  equation  consists,  then,  in 
bringing  the  unknown  quantity  by  itself,  on  one  side,  and  all 
the  known  quantities  on  the  other  side,  without  destroying 
the  equation. 

To  effect  this,  it  is  evident  that  one  of  the  members  must 
be  as  much  increased  or  diminished  as  the  other.  If  a  quan- 
tity be  added  to  one,  and  not  to  the  other,  the  equality  will 
be  destroyed. 

But  the  members  will  remain  equal ; 
If  the  same  or  equal  quantities  be  added  to  each.     Ax.  1. 
If  the  same  or  equal  quantities  be  subtracted  from  each.  Ax.  2. 
If  each  be  multiplied  by  the  same  or  equal  quantities.  Ax.  3. 
If  each  be  divided  by  the  same  or  equal  quantities.     Ax.  4. 

177.  It  may  be  farther  observed  that,  in  general,  if  the 
unknown  quantity  is  connected  with  others  by  addition,  mul- 
tiplication, division,  &c.  the  reduction  is  made  by  a  contrary 
process.  If  a  known  quantity  is  added  to  the  unknown,  the 
equation  is  reduced  by  subtraction.     Jf  one  is  multiplied  by 


88  SIMPLE      EaUATIONS. 

the  other,  the  reduction  is  effected  by  division,  &c.  The 
reason  of  this  will  be  seen,  by  attending  to  the  several  cases 
in  the  following  articles. 

The  known  quantities  may  be  expressed  either  by  letters 
or  figures.  The  unknown  quantity  is  represented  by  one  of 
the  last  letters  of  the  alphabet,  generally,  a:,  y,  or  z.    (Art.  24.) 

The  principal  reductions  to  be  considered  in  this  section, 
are  those  which  are  effected  by  transposition,  multiplication, 
and  division.  These  ought  to  be  made  perfectly  familiar,  a3 
one  or  more  of  them  will  be  necessary,  in  the  resolution  of 
almost  every  equation. 

Transposition. 

178.  In  the  equation 

#-7=9, 

the  number  7  being  connected  with  the  unknown  quantity  x 
by  the  sign  — ,  the  one  is  subtracted  from  the  other.  To  re- 
duce the  equation  by  a  contrary  process,  let  7  be  added  to 
both  sides.     It  then  becomes 

#-7+7=9+7. 

The  equality  of  the  members  is  preserved,  because  one  is 
as  much  increased  as  the  other.  (Axiom  1.)  But  on  one 
side,  we  have  —7  and  +7.  As  these  are  equal,  and  have 
contrary  signs,  they  balance  each  other,  and  may  be  cancel- 
led.    (Art.  70.)     The  equation  will  then  be 

#=9+7. 

Here  the  value  of  x  is  found.  It  is  shown  to  be  equal  to 
9+7,  that  is  to  16.  The  equation  is  therefore  reduced. 
The  unknown  quantity  is  on  one  side  by  itself,  and  all  the 
known  quantities  on  the  other  side. 

In  the  same  manner,  if  x— b=a 

Adding  b  to  both  sides  x—b-\-b=a+b 

And  cancelling  (—b+b)  x=a+b. 

Here  it  will  be  seen  that  the  last  equation  is  the  same  as 
the  first,  except  that  b  is  on  the  opposite  side,  with  a  contrary 
sign. 

Next  suppose  y+c=d. 


SIMPLE      EaUATIONS.  89 

Here  c  is  added  to  the  unknown  quantity  y.  To  reduce  the 
equation  by  a  contrary  process,  let  c  be  subtracted  from  both 
sides,  that  is,  let  —  c,  be  applied  to  both  sides.     We  then  have 

y+c—c=d—c. 

The  equality  of  the  members  is  not  affected;  because  one 
is  as  much  diminished  as  the  other.  When  (+c—c)  is  can- 
celled, the  equation  is  reduced,  and  is 

y=d-c. 

This  is  the  same  as  y+c=d,  except  that  c  has  been  trans- 
posed, and  has  received  a  contrary  sign.  We  hence  obtain 
the  following  general  rule : 

When  known  quantities  are  connected  with  the  unknown 
quantity  by  the  sign  +  or  — ,  the  equation  is  reduced  by 
transposing  the  known  quantities  to  the  other  side,  and  pre- 
fixing the  contrary  sign. 

This  is  called  reducing  an  equation  by  addition  or  subtrac- 
tion, because  it  is,  in  effect,  adding  or  subtracting  certain 
quantities,  to  or  from  each  of  the  members. 

Ex.  1.  Reduce  the  equation  x+3b—m=-h—d 

Transposing  +36,  we  have  x—m=h—d  —3b 

And  transposing  —m,  x=  h  —d—3b+m. 

179.  When  several  terms  on  the  same  side  of  an  equa- 
tion are  alike,  they  may  be  united  in  one,  by  the  rules  for  re- 
duction in  addition.     (Art.  67  and  71.) 

Ex.  2.  Reduce  the  equation  x-\-5b— 4h=7b 

Transposing  5b  —  Ah  x=7b—5b+4h 

Uniting  lb— 5b  in  one  term  x=2b+4h. 

180.  The  unknown  quantity  must  also  be  transposed, 
whenever  it  is  on  both  sides  of  the  equation.  It  is  not  ma- 
terial on  which  side  it  is  finally  placed.  For  if  x=3,  it  is 
evident  that  3=x.  It  may  be  well,  however,  to  bring  it  on 
that  side,  where  it  will  have  the  affirmative  sign,  when  the 
equation  is  reduced. 

Ex.  3.  Reduce  the  equation  2x+2h=h+d+3x 

By  transposition  2h—h—d—3x—2x 

And  h—d=x 

8* 


90  SIMPLE      EQUATIONS. 

181.  When  the  same  term,  with  the  same  sign,  is  on 
opposite  sides  of  the  equation,  instead  of  transposing,  we 
may  expunge  it  from  each.  For  this  is  only  subtracting  the 
same  quantity  from  equal  quantities.     (Ax.  2.)   . 

Ex.  4.   Reduce  the  equation  x+3h+d=b+3h+7d 

Expunging  3h  x+d=b+ld 

And  x=b+6d. 

1  8£.  As  all  the  terms  of  an  equation  may  be  transposed, 
or  supposed  to  be  transposed ;  and  it  is  immaterial  which 
member  is  written  first ;  it  is  evident  that  the  signs  of  all  the 
terms  may  be  changed,  without  affecting  the  equality. 

Thus,  if  we  have  x—b—d—a 

Then  by  transposition*  —  d+a=  —  x-\-b 

Or,  inverting  the  members  —  x-\-b=  —  d+a. 

183.  If  all  the  terms  on  one  side  of  an  equation  be  trans- 
posed, each  member  will  be  equal  to  0. 

Thus,  if  x+b=d,  then    x+b— d=0. 

It  is  frequently  convenient  to  reduce  an  equation  to  this 
form,  in  which  the  positive  and  negative  terms  balance  each 
other.  In  the  example  just  given,  x-\-b  is  balanced  by  —  d. 
For  in  the  first  of  the  two  equations,  x+b  is  equal  to  d. 

Ex.  5.  Reduce  3—2n+x=h  —  5+2x—2n. 

6.  Reduce  3x— n2  +ah=n+2x— ah  —  n2. 

7.  Reduce  a-3x+12=b-12-4x-a2. 

8.  Reduce  14— y+ab+c2  =  10  —  ab— c— 2y. 

Reduction  of  Equations  by  Multiplication. 

1 84.  The  unknown  quantity,  instead  of  being  connected 

with  a  known  quantity  by  the  sign  +  or  —  ,  may  be  divided 

x 
by  it,  as  in  the  equation    -  =  b. 

Here  the  reduction  can  not  be  made,  as  in  the  preceding 
instances,  by  transposition.  But  if  both  members  be  multi- 
plied by  a,  (Art.  176.)  the  equation  will  become, 

x~ab.. 


SIMPLE      EQUATIONS.  91 

For  a  fraction  is  multiplied  into  its  denominator,  by  re- 
moving the  denominator.     This  has  been  proved  from  the 
Eroperties  of  fractions.     (Art.  162.) 
lence, 

When  the  unknown  quantity  is  divided  by  a  known  quan- 
tity, the  equation  is  reduced  by  multiplying  each  side  by  this 
known  quantity. 

It  must  be  observed,  that  every  term  of  the  equation  is  to 
be  multiplied.  For  the  several  terms  in  each  member  consti- 
tute a  compound  multiplicand,  which  is  to  be  multiplied  ac- 
cording to  Art.  95. 

The  same  transpositions  are  to  be  made  in  this  case,  as  in 
the  preceding  examples. 

x 
Ex.  1.  Reduce  the  equation  — [-a=b+d 


c 


Multiplying  both  sides  by 


The  product  is  x-\-ac=bc-{-cd 

m         And  x=bc-\-cd—ac. 

x — 4 

2.  Reduce  the  equation  — • — }-5=20. 

Ans.     #=94. 

x 

3.  Reduce  the  equation  ~~TX"^=^* 

Ans.  x=ah+bh—ad—bd. 

1  8«>.  When  the  unknown  quantity  is  in  the  denominator 
of  a  fraction,  the  reduction  is  made  in  a  similar  manner,  by 
multiplying  the  equation  by  this  denominator. 

Ex.  4.  Reduce  the  equation  +7—8. 

Multiplying  by  10-a;  6+70-7x=80-8x. 

And  x=4. 

180.  Though  it  is  not  generally  necessary,  yet  it  is  often 
convenient,  to  remove  the  denominator  from  a  fraction  con- 
sisting of  known  quantities  only.  This  may  be  done,  in  the 
same  manner,  as  the  denominator  is  removed  from  a  fraction, 
which  contains  the  unknown  quantity. 


92  SIMPLE      EaUATIONS. 

x     d     h 

Take  for  example  -  =  r  +  - 

r  a     b      c 

ad     ah 

Multiplying  by  a  x  =  -r  +  — 

abh 
Multiplying  by  b  -  bx=ad-\ 

Multiplying  by  c  bcx  =  acd  +  abh. 

Or  we  may  multiply  by  the  product  of  all  the  denomina- 
tors at  once. 

x      d     h 
In  the  same  equation  -  =  r  +  - 

obex      abed      abch 
Mutiplying  by  abc  ==  —, 1 

Then  by  cancelling  from,  each  term,  the  letter  which  is 
common  to  its  numerator  and  denominator,  (Art.  149.)  we 
have         bcx=acd+abh,    as  before. 

Hence, 

An  equation  may  be  cleared  of  fractions  by  multiplying 
each  side  into  all  the  denominators. 

Thus  the  equation 

is  the  same  as  dgmx  =  abgm  +  adem  —  adgh. 

And  the  equation 

is  the  same  as  30a:  =  40  +  48  +  180. 

187.  An  equation  will  be  cleared  of  fractions,  if  each 
member  be  multiplied  by  any  common  multiple  of  the  de- 
nominators, that  is,  by  any  quantity  which  is  divisible  by  all 
the  denominators.  For  the  numerator  of  each  fraction  will 
be  multiplied  into  this  quantity,  (Art.  161.)  and  will  thus  be- 
come divisible  by  the  denominator. 

The  product  of  all  the  denominators  is  of  course  a  multi- 
ple of  them;  but  often  it  is  not  their  least  common  multiple, 
and  then  it  is  convenient,  in  clearing  the  equation  of  frac- 


X 

a 

b       e 
=  77+- 

m 

dgmx- 

=  abgm  ■ 

■\-adem 

X 

2  = 

2      4 
=  3  +  5 

A 

SIMPLE      EQUATIONS.  93 

tions,  to  multiply  by  the  least  common  multiple  of  the  denom- 
inators, instead  of  their  product. 

Thus,  in  the  preceding  example,  it  is  sufficient  to  multiply  by 
30,  instead  of  60,  which  is  the  product  of  the  denominators. 

In  clearing  an  equation  of  fractions,  it  will  be  necessary  to 
observe,  that  the  sign  —  prefixed  to  any  fraction,  denotes  that 
the  whole  value  is  to  be  subtracted,  (Art.  147.)  which  is  done 
by  changing  the  signs  of  all  the  terms  in  the  numerator. 

^  .  a—d  3b—2hm—6n 

The  equation  =  c 

^  x  r 

is  the  same  as         ar—dr=  crx—3bx+2hmx-\~6nx. 
Reduction  of  Equations  by  Division. 

188.  When  the  unknown  quantity  is  multiplied  into  any 
known  quantity,  the  equation  is  reduced  by  dividing  both  sides 
by  this  known  quantity.     (Ax.  4.) 

Ex.  1.  Reduce  the  equation  ax+b—3h=d 

By  transposition  ax—d-\-3h—b 

Dividing  by  a 


d+Sh-b 


a 


2.  Reduce  the  equation  2x= -r  +  4b 


Ans. 


c      h 
ah—cd-\-4bch 
2cli  ' 


1 89.  If  the  unknown  quantity  has  co-efficients  in  several 
terms,  the  equation  must  be  divided  by  all  these  co-efficients, 
connected  by  their  signs,  according  to  Art.  124. 

Ex.  3.  Reduce  the  equation  3x—bx=a—d 

That  is,  (Art.  119.)  (3  —  b)x=a— d 

a—d 


Dividing  by  3  —  b 
Reduce  the  equati 
Dividing  by  a-f  1 


x=- 


3-b 
Reduce  the  equation  ax+x=h— 4 

h-4 

x= 

a+l 


94  SIMPLE      EaUATIONS. 

x—b      a+d 


5.  Reduce  the  equation 


Ans.  x= 


4 
ah-\-dh—\b 


4h-4 


190.  If  any  quantity,  either  known  or  unknown,  is 
found  as  a  factor  in  every  term,  the  equation  may  be  divided 
by  it.  On  the  other  hand,  if  any  quantity  is  a  divisor  in 
every  term,  the  equation  may  be  multiplied  by  it.  In  this 
way,  the  factor  or  divisor  will  be  removed,  so  as  to  render 
the  expression  more  simple. 

Ex.  6.  Reduce  the  equation  ax+Sab=6ad+a 

Dividing  by  a  x+3b=6d+l 

And  x=6d+l-3k 


7.  Reduce  the  equation 


x+l      b      h—d 


xxx 
Ans.  x=h—d+b—\. 
8.  Reduce  the  equation  x(a+b)— a— b=d(a+b) 

Dividing  by  a+b  (Art.  119  and  123.)  x—  l=d 

And  x=d+l. 

191.  If  for  any  term  or  terms  in  an  equation,  any  other 
expression  dF  the  same  value  be  substituted,  it  is' manifest 
that  the  equality  of  the  sides  will  not  be  affected. 

64 

Thus,  instead  of  16,  we  may  write  2X8,  or  —  >  or  25— 9,&c. 

For  these  are  only  different  forms  of  expression  for  the 
same  quantity. 

192.  It  will  generally  be  well  to  have  the  several  steps, 
in  the  reduction  of  equations,  succeed  each  other  in  the  fol- 
lowing order. 

First,  Clear  the  equation  of  fractions.     (Art.  186,  7.) 
Secondly,  Transpose  and  unite  the  terms.     (Art.  178,  &c.) 
Thirdly,  Divide  by  the  co- efficients  of  the  unknown  quan- 
tity.    (Arts.  188,  9.) 

It  is,  however,  sometimes  convenient  to  unite  terms  that 
can  be  united,  before  clearing  the  equation  of  fractions 


SIMPLE      EaUATIONS.  05 

Thus  the  following  equation  might  be  reduced,  by  first 
transposing  6  and  uniting  it  with  7,  and  then  proceeding  in 
the  usual  way. 

Examples. 

miX  5X 

1.  Reduce  the  equation  —  +  6  =  —  +7 

4  o 

Clearing  of  fractions  24#+192  =  20#+224 

Transp.  and  uniting  terms  4#  =  32 

Dividing  by  4  x  =  8. 

X  XX 

2.  Reduce  the  equation  -  +  A  =  r \-d 

Clearing  of  fractions  hcx+abx—acx  =  abcd—abch 

abcd—abch 
DividinS  x=te+aJ=a-c 

3.  Reduce  24+5#— 30=159-6#.  Ans.  #=15. 

„    ,         2#-7  x      #-13  .  166 

4.  Reduce  — 5  =  - —  •  Ans.  a^-jy 

3 

5.  Reduce   3#+8+  -#=7#-22. 

6.  Reduce   5f— -=2#— 16$. 

J--_  3#      7#      3#      Ix 

7.  Reduce   - --  +  T  =  ¥  -15. 

f         X         X  X 

8.  Reduce   -  +  -  -  -  =  7#--  -734. 

31  13  3 

9    Reduce    --#4-134-  —  #=ll#+-#+7£. 

«#-10      #      6-5  •  30-26 

10.  Reduce   -— -  =  -  -  —  Ans.  *= :^3 

#+3      #  A  3&(4a— 1) 

11.  Reduce    -^-=^  +  12.  Ans.  #=  — ^7— 


96  BlMfLE      EQUATIONS. 

12.  Keduce 

13.  Reduce 

14.  Reduce   ^ 

39 

_  a; 

15.  Reduce    -  —  -  =  -  — 


a: 

- 1  —  u. 

13-2-^i  =  11- 

a;      a: 
2~~3  = 

=a;-21. 

a;      x 

a:        5 

3~4: 

=  2""  12* 

2a;+5 
3 

X 

12=2- 

3-7 

4 

x+7 

2a:-7 

X          *r 

2 

3 

i~lb- 

6ar- 1 

7 

3a:— 8 
5 

16-a; 
2 

-l+^-. 

2a;— 3 

4a;— 9 

a;-l 

7 

16.  Reduce 

17.  Reduce 

18.  Reduce 

19.  Reduce 

20.  Reduce    (a— x)(b+x)-—  b(a— c)+x2  =  ^- 


4  5a;-6  2  12 

a2c 


An,  *=^> 

0 

21.  Reduce    — —  :  5  ; :  — ^—  :  12.     See  Art  193. 

O  o 

103.  Sometimes  the  conditions  of  a  problem  are  at  first 
stated,  not  in  an  equation,  but  by  means  of  a  proportion. 
To  show  how  this  may  be  reduced  to  an  equation,  it  will  be 
necessary  to  anticipate  the  subject  of  a  future  section,  so 
far  as  to  admit  the  principle  that  "when  four  quantities  are 
in  proportion,  the  product  of  the  two  extremes  is  equal  to  the 
product  of  the  two  means  ;"  a  principle  which  is  at  the  found- 
ation of  the  Rule  of  Three  in  arithmetic.     See  Arithmetic. 


Thus,  if  alb  I 

:  c :  d, 

then  ad=bc. 

And  if      3  :  4-! 

:  6 : 8, 

then  3X8=4X6, 

Hence, 

A  proportion  is  converted  into  an  equation  by  making  the 
product  of  the  extremes,  one  side  of  the  equation;  and  tlie 
product  of  the  means ,  the  other  side. 


SIMPLE      EQUATIONS.  97 

Ex.  1.  Reduce  to  an  equation  ax  '.  b  \  \  ch  \  d. 

The  product  of  the  extremes  is  adx 

The  product  of  the  means  is  bch 

,  The  equation  is,  therefore  adx=bch. 

2.  Reduce  to  an  equation  a+b  '.  c  '.  '.  h— m  '.  y. 

The  equation  is  ay+by=ch—cm. 

194:*  On  the  other  hand,  an  equation  may  be  converted 
into  a  proportion,  by  resolving  one  side  of  the  equation  into 
two  factors,  for  the  middle  terms  of  the  proportion  ;  and  the 
other  side  into  two  factors,  for  the  extremes. 

As  a  quantity  may  often  be  resolved  into  different  pairs  of 
factors  (Art.  30.)  ;  a  variety  of  proportions  may  frequently 
be  derived  from  the  same  equation. 

Ex.  1.  Reduce  to  a  proportion  abc=deh. 

The  side  abc  may  be  resolved  into*aXbc,  or  abXc,  or  acXb. 

And  deh  may  be  resolved  into         dXeh,  or  deXh,  or  dhXe. 

Therefore    a\  d  Weh'.bc  And  ac  '.  dh\\  e    \b 

And  ab  '.  de\\h    '.  c  And  ac  '.  d    \\eh\b, &e. 

For  in  each  of  these  instances,  the  product  of  the  extremes 
is   abc,  and  the  product  of  the  means   deh. 

2.  Reduce  to  a  proportion  ax+bx=cd—ch 

The  first  member  may  be  resolved  into       x(a+b) 
And  the  second  into  c(d—h) 

Therefore  x\  c\\  d—h  '.  a+b    And  d—h  \x\\  a+b  \  c,&c. 

Solution  op  Problems. 

195.  In  the  solution  of  problems,  by  means  of  equations, 
two  things  are  necessary :  First,  to  translate  the  statement 
of  the  question  from  common  to  algebraic  language,  in  such 
a  manner  as  to  form  an  equation :  Secondly,  to  reduce  this 
equation  to  a  state  in  which  the  unknown  quantity  will  stand 
by  itself,  and  its  yalue  be  given  in  known  terms,  on  the  oppo- 
site side.  The  manner  in  which  the  latter  is  effected,  has 
already  been  considered.     The  former  will  probably  occasion 

9 


98  SIMPLE      EQUATIONS. 

more  perplexity  to  a  beginner ;  because  the  conditions  of 
questions  are  so  various  in  their  nature,  that  the  proper  method 
of  stating  them  cannot  be  easily  learned,  like  the  reduction 
of  equations,  by  a  system  of  definite  rules.  Practice  will 
soon  remove  a  great  part  of  the  difficulty. 

The  following  rule  may  however  be  of  some  use. 

Represent  the  unknown  quantity  by  a  letter,  and  indicate 
by  algebraic  signs,  the  operations  that  would  be  required  to 
verify  the  value  of  the  unknown  quantity,  if  it  were  given. 

196.  It  is  one  of  the  principal  peculiarities  of  an  alge- 
braic solution,  that  the  quantity  sought  is  itself  introduced 
into  the  operation.  This  enables  us  to  make  a  statement  of 
the  conditions  in  the  same  form,  as  though  the  problem  were 
already  solved.  Nothing  then  remains  to  be  done,  but  to  re- 
duce the  equation,  and  to  find  the  aggregate  value  of  the 
known  quantities.  (Art.  48.)  As  these  are  equal  to  the 
unknown  quantity  on  the  other  side  of  the  equation,  the 
value  of  that  also  is  determined,  and  therefore  the  problem  is 
solved. 

Problem  1.  A  man  being  asked  how  much  he  gave  for  his 
watch,  replied :  If  you  multiply  the  price  by  4,  and  to  the 
product  add  70,  and  from  this  sum  subtract  50,  the  remainder 
will  be  equal  to  220  dollars. 

To  solve  this,  we  must  first  translate  the  conditions  of 
the  problem,  into  such  algebraic  expressions  as  will  form  an 
equation. 

The  value  of  the  unknown  quantity  will  hereafter  be  found 
to  be  50.     If  this  value  were  given,  we  should  verify  it  thus. 

Let  the  price  of  the  watch  be  50. 

This  price  is  to  be  multiplied  by  4,  which  makes  200 

To  the  product,  70  is  to  be  added,  making  200+70 

From  this,  50  is  to  be  subtracted,  making  200+70—50 

And  the  remainder  is  equal  to  220,  that  is,  200+70  —  50—220. 

Reducing  the  first  member,  we  see  that  this  is  a  true  equa- 
tion, and  hence  infer  that  50  is  the  true  value  of  the  unknown 
quantity. 

According  to  the  preceding  rule  then,  we  may  form  an 
equation  for  the  solution  of  the  problem,  as  follows. 


SIMPLE      ECIUATIONS.  99 

Let  the  price  of  the  watch  be  denoted  by  x. 
Multiplying  this  by  4  makes  4x 
Adding  70  to  the  product  makes   4x +70 
Subtracting  50  from  this  leaves   4x +70— 50. 

Here  we  have  a  number  of  the  conditions,  expressed  in 
algebraic  terms ;  but  have  as  yet  no  equation.  We  must 
observe  then,  that  by  the  last  condition  of  the  problem,  the 
preceding  terms  are  said  to  be  equal  to  220. 

We  have,  therefore,  this  equation         4#+70— 50=220 
Which  reduced  gives  x =50. 

Here  the  value  of  x  is  found  to  be  50  dollars,  which  is  the 
price  of  the  watch. 

197*  We  may  prove  whether  we  have  obtained  the  true 
value  of  the  letter  which  represents  the  unknown  quantity, 
by  substituting  this  value,  for  the  letter  itself,  in  the  equation 
which  contains  the  first  statement  of  the  conditions  of  the 
problem ;  and  seeing  whether  the  sides  are  equal,  after  the 
substitution  is  made.  For  if  the  answer  thus  satisfies  the 
conditions  proposed,  it  is  the  quantity  sought.  Thus,  ,in  the 
preceding  example, 

The  original  equation  is  4#+70— 50=220 

Substituting  50  for  x,  it  becomes       4X50+70  —  50=220 
That  is,  220=220. 

Prob.  2.  What  number  is  that,  to  which  if  its  half  be 
added,  and  from  the  sum  20  be  subtracted,  the  remainder  will 
be  a  fourth  of  the  number  itself? 

In  stating  questions  of  this  kind,  where  fractions  are  con- 

1  x 

cerned,  it  should  be  recollected,  that  ^x  is  the  same  as  - ; 

o  3 

2        2# 
that  -=x=  —  >  &c.     (Art.  164.) 
o         o 

In  this  problem,  let  x  be  put  for  the  number  required. 

x  x 

Then  by  the  conditions  proposed,  x+~  —20=- 

2  4 

And  reducing  the  equation  x=lG. 

Proof,  16+^-20=^. 

2  4 


100  SIMPLE      EQUATIONS. 

Prob.  3.  A  father  divides  his  estate  among  his  three  sons, 
in  such  a  manner,  that, 

The  first  has  $1000  less  than  half  of  the  whole; 
The  second  has  800  less  than  one  third  of  the  whole ; 
The  third  has  600  less  than  one  fourth  of  the  whole. 
What  is  the  value  of  the  estate  ? 

If  the  whole  estate  be  represented  by  x}  then  the  several 

X  XX 

shares  will  be    -  —1000,  and  -  —800,  and  -  —600. 
£  o  4 

And  as  these  constitute  the  whole  estate,  they  are  together 
equal  to  x. 

X  XX 

We  have  then  this  equation  -  — 1000+ -  —800+  -  —  600=x. 

2  3  4 

Which  reduced  gives  #=28800. 

^      .  28800     „  28800  28800 

Proof,  — 1000+  — 800+  — 600=28800. 

&  S  4 

198.  Letters  may  be -employed  to  express  the  known 
quantities  in  an  equation,  as  well  as  the  unknown.  A  par- 
ticular value  is  assigned  to  the  letters,  when  they  are  intro- 
duced into  the  calculation ;  and  at  the  close,  the  numbers  are 
restored.     (Art.  47.) 

Prob.  4.  If  to  a  certain  number  720  be  added,  and  the 
sum  be  divided  by  125 ;  the  quotient  will  be  equal  to  7392 
divided  by  462.     What  is  that  number  ? 

Let  x=  the  number  required. 

a=720  tf=7392 

5=125  A=462. 

Then  by  the  conditions  of  the  problem 

Therefore  a;= 

^          .        .             .                 (125X7392) -(720X462)     __ 
Restoring  the  numbers,  x= — - =1280. 


SIMPLE      EQUATIONS.  101 

109.  To  avoid  an  unnecessary  introduction  of  unknown 

Siantities  into  an  equation,  it  may  be  well  to  observe,  in  this 
ace,  that  when  the  sum  or  difference  of  two  quantities  is 
given,  both  of  them  may  be  expressed  by  means  of  the  same 
letter.  For  if  one  of  the  two  quantities  be  subtracted  from 
their  sum,  it  is  evident  the  remainder  will  be  equal  to  the 
other.  And  if  the  difference  of  two  quantities  be  subtracted 
from  the  greater,  the  remainder  will  be  the  less. 

Thus  if  the  sum  of  two  numbers  be  20 

And  if  one  of  them  be  represented  by  x 

The  other  will  be  equal  to  20— x* 

Prob.  5.  What  two  numbers  are  those  whose  sum  is  50, 
and  whose  ratio  is  that  of  3  to  2  ? 

Let  the  larger  number  be  denoted  by    x 
Then  the  smaller  will  be  50— x. 

And  x  :  50— x : :  3  : 2 

Therefore  x =30,  the  greater 

And  50- #=20,  the  less. 

Prob.  6.  What  two  numbers  are  those  whose  sum  is  a, 
and  whose  ratio  is  that  of  m  to  n. 

Let  x  be  put  for  one  number 
Then  the  other  will  be  a—x. 
And  x  :  a—x  \\m\n 

am 


Hence  x= 


m+n 


This  is  a  general  problem,  of  which  the  preceding  is  only 
a  particular  example,  where  the  values  of  a,  m  and  n  are  50 
3  and  2. 

These  letters  may  stand  for  any  other  numbers;  and  in 

each  case,  the  formula  x= — j —   will  give  the  value  of  the 

m+n  & 

unknown  quantity :  so  that  the  solution  of  problem  6,  affords 
the  solution  of  an  infinite  number  of  particular  prbblems  sim- 
ilar to  problem  5. 

Prob.  7.  What  number  is  that,  the  double  of  which  ex- 
ceeds two  thirds  of  its  half  bv  45.  Ans.  27. 

9* 


102  SIMPLE      ECIUATIONS. 

Prob.  8.  A  farmer  sold  14  bushels  of  wheat  at  a  certain 
price ;  and  afterward  sold  22  bushels  at  the  same  rate,  re- 
ceiving 56  shillings  more  in  this  case  than  in  the  former. 
What  was  the  price  of  a  bushel  ?  Ans.  7  shillings. 

Prob.  9.  What  number  is  that  the  treble  of  which  is  as 
much  above  25,  as  its  half  is  below  38. 

Prob.  10.  A  person  bought  209  gallons  of  beer,  which  ex- 
actly filled  four  casks  ;  the  first  held  three  times  as  much  as 
the  second,  the  second  twice  as  much  as  the  third,  and  the  third 
twice  as  much  as  the  fourth.  How  many  gallons  did  each 
hold?  Ans.   132,  44,  22,  and  11  gallons. 

Prob.  11.  Divide  75  into  two  such  parts,  that  the  greater 
being  divided  by  5,  and  the*  less  by  10,  the  difference  of  the 
quotients  shall  be  6. 

Prob.  12.  A  silversmith  has  three  pieces  of  metal,  which 
together  weigh  62  ounces.  The  second  weighs  5  ounces 
more  than  the  first,  and  the  third  7  ounces  more  than  the 
second.     What  are  their  weights  ? 

Ans.  15,  20  and  27  ounces. 

This  problem  may  be  generalized,  as  follows. 

Prob.  13.  The  sum  of  three  numbers  is  a;  the  second  ex- 
ceeds the  first  by  m,  and  the  third  exceeds  the  second  by  w. 
What  are  the  numbers  ? 

A         a—2m—n     a+m—n     a-Jrm-\-2n 

Ans.  — _ __,    _ — 

Prob.  14.  What  number  is  less  by  6,  than  the  sum  of  its 
naif,  its  third  and  its  fourth.  Ans.  72. 

Prob.  15.  Two  workmen  received  the  same  sum  for  their 
labor ;  but  if  one  had  received  12  dollars  more,  and  the  other 
3  dollars  less,  the  former  would  have  had  4  times  as  much 
as  the  latter.     What  did  they  receive  ? 

Ans.  8  dollars  each. 
Prob.  16.    (The  preceding  problem  generalized.)     What 
number  is  that  which,  when  increased  by  a,  is  m  times  as 
great  as  when  diminished  by  b.  a+mb 

Ans.  -. 

m—  1 

Prob.  17.  A  fortress  is  garrisoned  by  1800  men  ;  and  there 
are  7  times  as  many  infantry,  and  4  times  as  many  artillery 
as  cavalry.     How  many  are  there  of  each  ? 

Ans.  1050  infantry,  600  artillery  and  150  cavalry. 


SIMPLE      EaUATIONS.  103 

Prob.  18.  Divide  the  number  a  into  three  such  parts,  that 
the  first  shall  be  m  times,  and  the  second  n  times,  as  great  as 
the  third.  .  ma  na  a 

m+n+1     rn-\-n-\-\y    m+n+l 

Prob.  ,19.  A  farm  of  660  acres  is  divided  between  three 
persons.  C  has  as  many  acres  as  A  and  B  together,  and  the 
portions  of  A  and  B  are  in  the  ratio  of  4  to  7.  How  many 
acres  has  each  ? 

Prob.  20.  The  ingredients  of  a  loaf  of  bread  are  rice, 
flour  and  water;  and  the  weight  of  the  whole  is  18  pounds. 
The  weight  of  the  rice  increased  by  2  pounds  is  one  fifth  of 
the  weight  of  the  flour,  and  the  weight  of  the  wTater  is  one 
eighth  of  the  weight  of  the  flour  and  rice  together.  What 
is  the  weight  of  each  ? 

200*  Though  it  is  usual  to  begin  the  solution  of  a  prob- 
lem, by  assuming  a  letter  to  represent  some  unknown  quan- 
tity, it  is  sometimes  best  to  adopt  a  different  method. 

Thus  in  the  following  example,  where  three  numbers  are 
required,  which  are  as  2,  3  and  5,  they  may  be  represented 
by  2x,  3x  and  5x ;  x  being  put  for  half  'of  the  first  number. 
The  value  of  x  may  then  be  found,  and  the  required  numbers 
obtained,  by  multiplying  this  value  into  2,  3  and  5. 

Prob.  21.  There  are  three  pieces  of  cloth,  whose  lengths 
are  as  2,  3  and  5 ;  and  12  yards  being  cut  off  from  each,  the 
whole  quantity  is  diminished  in  the  ratio  of  10  to  7.  What 
was  the  length  of  each  piece  at  first  ? 

Prob.  22.  What  two  numbers  are  those,  whose  difference 
is  21,  and  whose  sum  is  greater  by  9  than  three  times  the 
less? 

Prob.  23.  A  company  of  men  performed  a  piece  of  work 
in  6  days.  If  there  had  been  5  more,  they  would  have  fin- 
ished the  work  in  4  days.     Of  how  many  did  the  company 

consist  ? 

Prob.  24.  In  three  towns,  A,  B  and  C,  the  whole  number 
of  inhabitants  is  4600.  The  numbers  in  A  and  B  are  as  2 
to  3,  and  in  B  and  C  as  5  to  7.  What  is  the  number  in  each 
of  the  towns  ? 

Prob.  25.  A  certain  sum  is  divided  between  three  persons  ; 
A  receives  1000  dollars  less  than  the  half,  B  J*00  dollars  more 


104  SIMPLE      EQUATIONS. 

than  the  third,  and  C  300  dollars  more  than  the  fourth  of  the 
whole.     How  much  does  each  receive  ? 

Prob.  26.  A  trader  first  lost  one  third  of  his  capital,  and 
then  gained  400  dollars ;  after  which  he  lost  one  fourth  of 
what  he  then  had,  and  gained  200  dollars ;  lastly  he  lost  one 
fifth  of  what  he  then  had,  and  saved  1600  dollars^  What 
had  he  at  first  ? 

Prob.  27.  Divide  103  into  four  such  parts,  that,  the  first 
increased  by  7,  the  second  diminished  by  5,  the  third  multi- 
plied by  4,  and  the  fourth  divided  by  3,  shall  all  be  equal. 

Prob.  28.  A  person  has  two  pieces  of  land,  A  and  B,  the 
former  of  which  is  three  fourths  as  large  as  the  latter.  Hav- 
ing sold  3  acres  less  than  half  of  A,  and  12  acres  more  than 
two  thirds  of  B,  he  finds  that  the  remaining  part  of  B  is  only 
two  fifths  as  large  as  that  of  A.  What  was  the  size  of  each 
field  at  first  ? 

Prob.  29.  A  person  has  two  kinds  of  sugar,  one  worth  12 
cents,  and  the  other  7  cents  a  pound.  How  much  of  each 
must  he  take,  to  make  a  mixture  worth  10  cents  a  pound? 

Prob.  30.  A  person  has  two  kinds  of  sugar,  one  worth  m 
cents  a  pound,  and  the  other  n  cents.  How  much  of  each 
must  he  take  to  make  a  mixture  worth  r  cents  a  pound  ? 

Prob.  31.  A  reservoir  containing  765  gallons  of  water, 
was  emptied  by  two  buckets  in  36  minutes.  The  smaller 
bucket  holding  two  thirds  as  much  as  the  other,  was  emptied 
once  a  minute,  and  the  larger  one,  three  times  in  four  min- 
utes.    What  was  the  size  of  each  ? 

Prob.  32.  A  man  and  his  family  consume  a  barrel  of  flour 
in  50  days.  When  he  is  absent,  the  barrel  lasts  10  days  longer. 
How  long  would  it  last,  if  he  used  it  alone  ? 

Prob.  33.  A  man  and  his  family  consume  a  barrel  of  flour 
in  m  days.  When  he  is  absent,  the  barrel  lasts  n  days  lon- 
ger.    How  long  would  it  last,  if  he  used  it  alone  ? 

m.(m+n) 
Ans.  — ■ days. 

Prob.  34.  A  merchant  bought  two  kinds  of  wine  in  equal 
quantities;  giving  for  one  4  shillings  a  gallon,  and  for  the  other 
7.  By  mixing  them  and  selling  the  mixture  at  6  shillings  a 
gallon,  he  gained  18  shillings.  How  many  gallons  of  each 
did  he  buy  ? 


SIMPLE      EaUATIONS.  105 

Prob.  35.  A  merchant  bought  two  kinds  of  wine  in  equal 
quantities,  giving  for  one  m  shillings  a  gallon,  and  for  the 
other  n  shillings  a  gallon.  By  mixing  them 'and  selling  the 
mixture  at  r  shillings  a  gallon,  he  gained  a  shillings.  How 
many  gallons  of  each  did  he  buy  ?  a 

™  '  2? — m—n 

Prob.  36.  What  number  is  that,  to  which  if  2,  5  and  11  be 
severally  added,  the  first  sum  shall  be  to  the  second,  as  the 
second  to  the  third? 

Prob.  37.  What  number  is  that,  to  which  if  a,  b  and  c  be 

severally  added,  the  first  sum  shall  be  to  the  second,  as  the 

second  to  the  third?  b*—ac 

Ans.  — rr- 

'  .  a+c— 2b 

Prob.  38.  A,  B  and  C  make  a  join*-  stock.     A  puts  in  300 
/dollars  more  than  B,  and  200  dollars  less  than  C ;  and  the 
sum  of  the  shares  of  A  and  B,  is  to  the  sum  of  the  shares  of 
B  and  C  as  7  to  9.     What  did  each  put  in  ? 

Prob.  39.  A,  B  and  C  make  a  joint  stock.  A  puts  in  m 
dollars  more  than  B,  and  n  dollars  less  than  C ;  and  the  sum 
of  the  shares  of  A  and  B  is  to  the  sum  of  the  shares  of  B 
and  C,  as  p  to  q.     What  did  each  put  in  ? 

p(n—m)  +  qm     p(n-\-m)  —  qm     q(2n+m)—p(n-{-m) 

'~Wzp)  2(?-p)    '  :      Mq-p) 

Prob.  40.  A  laborer  is  paid^30  dollars  for  reaping  26  acres 
of  wheat  and  rye,  at  the  rate  of  a  dollar  an  acre  for  the  rye, 
and  a  dollar  and  a  quarter  for  the  wheat.  How  much  would 
he  have  received  for  reaping  the  wheat  alone  ? 

'    Prob.  41.  Divide  15  into  two  such  parts  that  the  difference 
of  their  squares  shall  be  105. 

Prob.  42.  Divide  the  number  a  into  two  such  parts  that 
the  difference  of  their  squares  shall  be  d. 

a2+d    a2-d 


Ans. 


2a  2a 


Prob.  43.  Out  of  a  certain  sum  of  money  a  man  paid  his 
creditors  432  dollars ;  one  third  of  the  remainder  he  lent  his 
friend  ;  he  then  spent  one  fourth  of  what  still  remained  ;  after 
which  he  had  one  fifth  of  the  money  left  ?  How  much  had 
he  at  first  ? 


106  SIMPLE      EaUATIONS. 

Prob.  44.  A  person  being  asked  the  hour,  answered  that  it 
was  between  two  and  three,  and  the  hour  and  minute  hands 
were  together.     What  was  the  time  ? 

Prob.  45.  A  merchant  buys  a  cask  of  wine  for  150  dollars, 
and  sells  5  gallons  less  than  two  thirds  of  the  whole  at  a  profit 
of  50  per  cent.  He  afterwards  sells  the  remainder  at  a  profit 
of  125  per  cent,  and  finds  that  he  has  cleared  80  per  cent, 
by  the  whole  transaction.  How  many  gallons  does  the  cask 
contain  ? 

Prob.  46.  In  a  naval  engagement  the .  number  of  ships 
taken  was  one  fifth  of  the  whole ;  the  number  burnt  was  1 
less,  and  the  number  sunk  2  more  than  the  number  taken ; 
and  13  escaped.     Of  how  many  did  the  fleet  consist? 

Prob.  47.  The  yearly  rent  of  a  farm  is  80  dollars  in  mo- 
ney, and  a  certain  quantity  of  wheat.  When  wheat  is  worth 
a  dollar  a  bushel,  the  rent  is  8  dollars  per  acre ;  but  when  it 
is  worth  a  dollar  and  a  quarter,  the  rent  per  acre  is  9  dollars. 
Of  how  many  acres  does  the  farm  consist  ? 

Prob.  48.  The  yearly  rent  of  a  farm  is  a  dollars  in  money, 
and  a  certain  quantity  of  wheat.  When  wheat  is  worth  m 
dollars  a  bushel,  the  rent  is  p  dollars  per  acre;  but  when  it 
is  worth  n  dollars,  the  rent  per  acre  is  q  dollars.  Of  how 
many  acres  does  the  farm  consist  ?  (n—m)a 

np—mq 

Prob.  49.  A  and  B  are  traveling  in  the  same  direction, 
and  A  is  15  miles  in  advance  of  B.  If  B  goes  five  miles 
while  A  goes  four,  how  far  must  he  travel  to  overtake  A  ? 

Prob.  50.  A  and  B  are  traveling  in  the  same  direction, 
and  A  is  d  miles  in  advance  of  B.  If  B  goes  m  miles  while 
A  goes  n,  how  far  must  he  travel  to  overtake  A  ? 

md 

Ans. 

m—n 


SIMPLE      EQUATIONS,     AC.  107 


SECTION    VIII. 

SIMPLE    EQUATIONS    CONTAINING    TWO    OR    MORE 
UNKNOWN    QUANTITIES. 

Art.  201.  In  the  examples  which  have  been  given  of 
the  resolution  of  equations,  in  the  preceding  sections,  each 
problem  has  contained  only  one  unknown  quantity.  Or  if, 
in  some  instances,  there  have  been  two,  they  have  been  so 
related  to  each  other,  that  they  have  both  been  expressed  by 
means  of  the  same  letter.     (Art.  199.) 

But  cases  frequently  occur,  in  which  several  unknown 
quantities  are  introduced  into  the  same  calculation.  And  if 
the  problem  is  of  such  a  nature  as  to  admit  of  a  determinate 
answer,  there  will  arise  from  the  conditions,  as  many  equa- 
tions independent  of  each  other,  as  there  are  unknown 
quantities. 

Equations  are  said  to  be  independent,  when  they  express 
different  conditions ;  and  dependent,  when  they  express  the 
same  conditions  under  different  forms.  The  former  are  not 
convertible  into  each  other.  But  the  latter  may  be  changed 
from  one  form  to  the  other,  by  the  methods  of  reduction 
which  have  been  considered.  Thus  b—x=y,  and  b=y+x, 
are  dependent  equations,  because  one  is  formed  from  the 
other  by  merely  transposing  x. 

In  solving  a  problem,  it  is  necessary  first  to  find  the  value 
of  one  of  the  unknown  quantities,  and  then  of  the  others  in 
succession.  To  do  this,  we  must  derive  from  the  equations 
which  are  given,  a  new  equation,  from  which  all  the  unknown 
quantities  except  one  shall  be  excluded. 

Solution  of  Problems  which  contain  Two  Unknown 
Quantities. 

202*  Thsre  are  three  methods  employed  in  eliminating 
unknown  quantities,  called 
Elimination  by  comparison, 
Elimination  by  substitution,  and 
Elimination  by  addition  and  subtraction. 


108  SIMPLE      EdUATIONS 

Elimination  by  Comparison. 
303.  Suppose  the  following  equations  are  given. 

1.  x+y=l4= 

2.  x— y=2. 

If  y  be  transposed  in  each,  they  will  become 

1.  x=14— y 

2.  x=2  +y. 

Here  the  first  member  of  each  of  the  equations  is  x,  and 
the  second  member  of  each  is  equal  to  x.  But  according  to 
axiom  11th,  quantities  which  are  respectively  equal  to  any 
other  quantity  are  equal  to  each  other ;  therefore, 

2+y=14— y. 
Here  we  have  a  new  equation,  which  contains  only  the 
unknown  quantity  y.     Hence, 

Rule  I.  To  eliminate  one  of  two  unknown  quantities,  and 
deduce  one  equation  from  two ; 

Find  the  value  of  one  of  the  unknown  quantities  in  each 
of  the  equations,  and  form  a  new  equation  by  making  one  of 
these  values  equal  to  the  other. 

That  quantity  which  is  the  least  involved  should  be  the 
one  which  is  chosen  to  be  eliminated. 

For  the  convenience  of  referring  to  different  parts  of  a  so- 
lution, the  several  steps  will,  in  future  be  numbered.  When 
an  equation  is  formed  from  one  immediately  preceding,  it  will 
be  unnecessary  to  specify  it.  In  other  cases,  the  number  of 
the  equation  or  equations  from  which  a  new  one  is  derived, 
will  be  referred  to. 

Prob.  1.  To  find  two  numbers  such,  that 
Their  sum  shall  be  24 ;  and 
The  greater  shall  be  equal  to  five  times  the  less. 
Let  #=the  greater;  And  y=the  less. 

1.  By  the  first  condition,  #+y=24 

2.  By  the  second,  x=5y 

3.  Transp.  y  in  the  first  equation,  xt=24  — y 

4.  Making  the  2d  and  3d  equal,   5y=24—y 

5.  And  y=4,  the  less  number. 


CONTAINING     TWO     UNKNOWN      aUANTITIES.      109 

Prob.  2.  To  find  two  numbers  such  that 

Their  difference  shall  be  a ;  and 
The  greater  shall  be  equal  to  m  times  the  less. 

ma  a 


Ans. 


m—\     771—1 


Prob,  3.  Given  ax—by—c  )  ^ '        ,  bd— c 


tx—by—c  ) 
x—  v—d  ) 


To  find  x.     Ans.  #=  - 


And        x—  y=d  )  '  *  6— a 

Elimination  by  Substitution. 

304:.  Suppose  the  equations  given  are,  as  in  the  former 
case, 

1.  o?+y=14 

2.  x—y=2. 

If  y  be  transposed,  the  first  equation  will  become, 

#=14— y. 

And  as  a?  is  here  equal  to  14— y,  we  may  in  the  second 
equation  substitute  this  value  of  x  instead  of  x  itself.  The 
second  equation  will  then  be  converted  into 

14-2t/=2. 

The  equality  of  the  two  sides  is  not  affected  by  this  altera- 
tion, because  we  only  change  one  quantity  x  for  another 
which  is  equal  to  it.  By  this  means  we  obtain  an  equation 
which  contains  only  one  unknown  quantity.     Hence, 

Rule  II.  To  eliminate  an  unknown  quantity, 

Find  the  value  of  one  of  the  unknown  quantities,  in  one  of 
the  equations;  and  then  in  the  other  equation  substitute 
this  value  for  the  unknown  quantity  itself 

Prob.  4.  A  privateer  in  chase  of  a  ship  20  miles  distant, 
sails  8  miles,  while  the  ship  sails  7.  How  far  must  the  pri- 
vateer sail  before  she  overtakes  the  ship  ? 

It  is  evident  that  the  whole  distance  which  the  privateer 
sails  during  the  chase,  must  be  to  the  distance  which  the  ship 
sails  in  the  same  time,  as  8  to  7. 
10 


110  SIMPLE      EQUATIONS 

Let   x—  the  distance  which  the  privateer  sails, 
And  y=  the  distance  which  the  ship  sails. 

1.  By  the  supposition,  z=*/+20  ) 

2.  And  also,  x  \  y  \  \  8  :  7  ) 

3.  Art.  193,  y=i^ 

4.  Substituting  %x  for  y,  in  the  1st  equation,  x=%x-\-20 

5.  Therefore,  x=160. 

Prob.  5.  What  fraction  is  that,  which"  becomes  two  thirds, 

when  2  is  added  to  its  numerator,   and  becomes  one  half, 

when  1  is  added  to  its  denominator.  8 

Ans.  -—• 
15 

Prob.  6.  There  are  two  numbers, 

Whose  sum  is  to  their  difference  as  9  to  4 ;  and 
The  greater  exceeds  twice  the  less  by  3. 

What  is  the  less  number  ?  Ans.  5. 

Elimination  by  Addition  and  Subtraction. 

205.  Suppose  the  given  equations  are 

x+3y=a 

2x—3y=b 

If  we  add  together  the  first  members  of  these  two  equa- 
tions, and  also  the  second  members,  we  shall  have 

3x=a+b, 

an  equation  which  contains  only  the  unknown  quantity  x. 
The  other,  having  equal  co-efficients  with  contrary  signs,  has 
disappeared.  (Art.  70.)  The  equality  of  the  sides  is  pre- 
served, because  we  have  only  added  equal  quantities  to  equal 
quantities. 

Again,  suppose  3x-\-y=h 

And  2x+y=d 

If  we  subtract  the  last  equation  from  the  first,  we  shall  have 

x-=h— d 

where  y  is  eliminated,  without  affecting  the  equality  of  the 
sides. 


CONTAINING     TWO     UNKNOWN      QUANTITIES.      Ill 

Again,  suppose  Sx— 2y=a  ) 
And  x+4y=b  ) 

Multiplying  the  1st  by  2,  6x— 4y = 2a 
Tljgn  adding  the  2d  and  3d,  7x=b  +2a. 

Again,  suppose  2x—4y=10 

And  x+Sy=S5 

Dividing  the  1st  by  2,  x—2y=5 

Then  subtracting  the  3d  from  the  2d,  5y=S0.     Hence, 

Rule  III.  To  eliminate  an  unknown  quantity, 

Multiply  or  divide  the  equations,  if  necessary,  in  such  a 
manner  that  the  term  which  contains  one  of  the  unknown 
quantities  shall  be  the  same  in  both. 

Then  subtract  one  equation  from  the  other,  if  the  signs 
of  this  unknown  quantity  are  alike,  or  add  them  together, 
if  the  signs  are  unlike. 

It  must  be  kept  in  mind  that  both  members  of  an  equation 
are  always  to  be  increased  or  diminished,  multiplied  or  divi- 
ded alike.     (Art.  176.) 

Prob.  7.  The  numbers  in  two  opposing  armies  are  such, 
that, 

The  sum  of  both  is  21110;  and 

Twice  the  number  in  the  greater  army,  added  to  three 
times  the  number  in  the  less,  is  52219. 

What  is  the  number  in  the  greater  army  ? 

Let  x—  the  greater.         And  y=  the  less. 

1.  By  the  first  condition,  x+y— 21110 

2.  By  the  second,  2#+3y=52219 

3.  Multiplying  the  1st  by  3,  3a;+3y,=63330 

4.  Subtracting  the  2d  from  the  3d,  x—  1 1 1 1 1 . 

Prob.  8.  Given  2x+3y=7,  and  Sx— 2y=4,  to  find  the 
value  of  x. 

Multiply  the  first  equation  by  2,  the  second  by  3,  and  add 
the  results.  Ans.  x=2* 


112  SIMPLE      EaUATIONS 

Prob.9.  Given  12#— 3y=30,  and  2x+5y=16,  to  find 
the  value  of  y. 

Divide  the  first  equation  by  3,  multiply  the  second  by  2, 
and  then  subtract  one  from  the  other.  Ans.  y=2. 

In  the  succeeding  problems,  either  of  the  three  rules  for 
eliminating  unknown  quantities  will  be  made  use  of,  as  will 
in  each  case  be  most  convenient. 

206.  When  one  of  the  unknown  quantities  is  determined, 
the  other  may  be  easily  obtained,  by  going  back  to  an  equa- 
tion which  contains  both,  and  substituting  instead  of  that 
which  is  already  found,  its  numerical  value. 

Prob.  10.  The  mast  of  a  ship  consists  of  two  parts; 

•One  third  of  the  lower  part  added  to  one  sixth  of  the  up- 
per part,  is  equal  to  28 ;  and, 

Five  times  the  lower  part,  diminished  by  six  times  the 
upper  part,  is  equal  to  12. 

What  is  the  height  of  the  mast  ? 

Let  x=  the  lower  part ;  And  y=  the  upper  part. 

1.  By  the  first  condition,  ^x+}y=  28 

2.  By  the  second,  5x—6y=   12 

3.  Multiplying  the  1st  by  6,  2x+  y=l6S 

4.  Dividing  the  2d  by  6,  £#—  y=2 

5.  Adding  the  3d  and  4th,  2z+fc=170 

6.  Therefore,  x =60,  the  lower  part. 
Then  by  the  3d  step,  2x+  y=168 

That  is,  substituting  60  for  x,  120+  y=16S 

Hence,  y=  108— 120=48,  the  upper  part. 


Prob.  11.  Given  |  +  |=2 
o      6 


>To  find  x  and  y. 


And 


3      4       6  J  Ans.  x=5,  y=6. 

Prob.  12.  Given  (a;+i)(y+2)  =  (a?-l)(y-2)+26 
And  5x—Sy=l 

To  find  the  values  of  x  and  y. 

Ans.  a: =5,  y=3. 


CONTAINING     TWO     UNKNOWN     ClUANTITIES.     118 

^    ,  „.         x—4   ■  y— x      10— y      19 

Prob.  13.  Given  _  +  ^r  +  _?  =  - 

.     x-f-y  ,  x+l      y—  1 

And    -7T  +  -2--    3    =3 

To  find  the  values  of  x  and  y. 

Ans.  #=5,  y=eT. 
JPro&.  14.  To  find  a;  and  y  from 

1.  The  equation 

2.  And 

3.  Multiplying  the  1st  by  3, 

4.  Multiplying  the  2d  by  5, 

5.  Subtracting  the  3d  from  the  4th, 

6.  Therefore 

7.  Substituting  3  for  x  in  the  2d, 

8.  Hence 

Prob.  15.  Given =m 

x    y 

Y  To  find  the  values  of  x  and  y. 

And =n 

x     y 

be— ad  be— ad 

Ans.  #=; r->    y= • 

on— dm     *     an— cm 

Prob.  16.  If  a  certain  number,  consisting  of  two  digits,  be 
divided  by  the  left  hand  digit  increased  by  1,  the  quotient 
will  be  9 ;  but  if  it  be  divided  by  the  right  hand  digit  dimin- 

9 
ished  by  1,  the  quotient  will  be  -•     What  is  the  number? 

Let  x=  the  left  hand  digit,  and  y=  the  right  hand  digit 
10* 


114  SIMPLE      EQUATIONS 

As  the  local  value  of  figures  increases  in  a  tenfold  ratio 
from  right  to  left,  the  required  number  =10£+y. 

Then  HSJT..9 

x  +1 

lOx+y     9 

And  ir=i=2; 

The  number  sought  is,  therefore,  27. 
Prob.  17.  If  a  certain  number  consisting  of  two  digits,  be 
divided  by  the  left  hand  digit  increased  by  m,   the  quotient 
will  be  a ;  but  if  it  be  divided  by  the  right  hand  digit  dimin- 
ished by  n,  the  quotient  will  be  b.     What  is  the  number  ? 

ab(\0m— n) 

S*  a+10b-ab' 

Prob.  18.  To  find  a  fraction  such  that, 

If  the  numerator  be  doubled,  and  the  denominator  be  di- 

7 
minished  by  2,  the  fraction  will  be  equal  to  - ;  but 

o 

If  the  denominator  be  doubled,  and  the  numerator  be  di- 
minished by  3,  the  fraction  will  be  equal  to 


6 


Ans.  1: 


Prob.  19.  What  two  numbers  are  those,  whose  sum  divided 
9 
by  the  greater  is  equal  to  -  >  and  whose  difference  is  less  than 

half  the  greater  by  6?  Ans.  20  and  16. 

Prob.  20.  What  two  numbers  are  those,  whose  sum  divided 
by  the  greater  is  equal  to  a,  and  whose  difference  is  less  than 
half  the  greater  by  b  ?  2b        2b(a—  1) 

Ans*  2^=3'      2a^T  ' 
Prob.  21.  If  B  gives  to  A  5  dollars,  A  will  then  have  half 
as  much  money  as  B.     But  if  A  gives  to  B  ten  dollars,  B 
will  then  have  five  times  as  much  as  A.     How  many  dollars 
has  each  ?  Ans.  A-has  25  and  B  65  dollars. 

Prob.  22.  A  merchant  mixes  wheat  flour  which  cost  him 
5  dollars  a  barrel,  with  rye  flour  which  cost  him  3  dollars  a 
barrel,  in  such  proportion  as  to  gain  33  J  per  cent,  by  selling 
the  mixture  at  6  dollars  a  barrel.  How  much  wheat  is  there 
in  40  barrels  of  the  mixture  ?  Ans.  30  barrels. 


CONTAINING    THREE    UNKNOWN     QUANTITIES.     115 

Prob.  23.  A  merchant  mixes  wheat  flour,  which  cost  him 
m  dollars  a  barrel,  with  rye  flour  which  cost  him  n  dollars  a 
barrel,  in  such  proportion  as  to  gain  p  per  cent,  by  selling 
the  mixture  at  r  dollars  a  barrel.  How  much  wheat  is  there 
in  a  barrels  of  the  mixture  ? 

(lOOr—  lOOn—  pn)a 

AnS-   Jm-n){lW+p) 

Prob.  24.  A  and  B  engage  to  reap  a  field  of  wheat  in  12 
days.  But  after  working  4  days,  A  is  called  off,  and  B  is  left 
to  finish  the  work,  which  he  does  in  20  days  more.  In  what' 
time  would  A  alone  reap  the  field  ?  Ans.  20  days. 

Prob.  25.  A  and  B  engage  to  reap  a  field  of  wheat  in  m 

days.     But  after  working  n  days,  A  is  called  off,  and  B  is  left 

to  finish  the  work,  which  he  does  in  r  days  more.     In  how 

many  days  would  A  alone  reap  the  field  ?  -^ 

mr 

Ans.  — ; 

r+n— m 

Solution    op   Problems    which   contain  Three  or   more 
Unknown  Quantities. 

30  7  •  In  the  examples  hitherto  given,  each  has  contained 
no  more  than  two  unknown  quantities.  And  two  indepen- 
dent equations  have  been  sufficient  to  express  the  conditions 
of  the  question.  But  problems  may  involve  three  or  more 
unknown  quantities ;  and  may  require  for  their  solution  as 
many  independent  equations. 

Suppose  x+  y+  z=12  } 

And         x-\-2y—2z—\0  >are  given,  to  find  x,  y  and  z. 

And         x+  y—  %=4    ) 

From  these  three  equations,  two  others  may  be  derived, 
which  shall  contain  only  two  unknown  quantities.  One  of 
the  three  in  the  original  equations  may  be  eliminated,  in  the 
same  manner  as  when  there  are,  at  first,  only  two,  by  the 
rules  in  Arts.  203,  4,  5. 

In  the  equations  given  above,  if  we  transpose  y  and  z,  we 
shall  have, 

Lin  the  first,  #=12—  y—  z  } 
In  the  second,  x=l0—2y+2z  > 
In  the  third,       #=  4—  y+  z  J 


116  SIMPLE      EQUATIONS 

From  these  we  may  deduce  two  new  equations,   from 
which  x  shall  be  excluded. 

By  making  the  1st  and  2d  equal,  12—  y—  2  =  10— 2y+2z  j 

By  making  the  2d  and  3d  equal,  10— 2y+2z=  4-  y+  z  I 
Reducing  the  first  of  these  two,  y=3z  — 2  ) 

Reducing  the  second^  y=  z+6  ) 

From  these  two  equations,  one  may  be  derived  containing 
only  one  unknown  quantity. 

Making  one  equal  to  the  other,  3%  — 2= z+6 

And  •  z=4.         Hence, 

To  solve  a  problem  containing  three  unknown  quantities, 
and  producing  three  independent  equations, 

First,  from  the  three  equations  deduce  two,  containing  only 
two  unknown  quantities. 

Then,  from  these  two  deduce  one,  containing  only  one  un- 
known quantity. 

For  making  these  reductions,  the  rules  already  given  are 
sufficient.     (Art.  203,  4,  5.) 

Prob.  26.  Let  there  be  given, 

1.  The  equation  #+5y+6%=53  } 

2.  And  x+3y-\-3z=30  >To  find  x,  y  and  z. 

3.  And  x+  y-f-  %=12  ) 

From  these  three  equations  to  derive  two,  containing  only 
two  unknown  quantities, 

4.  Subtract  the  2d  from  the  1st,     2y+3z=23 

5.  Subtract  the  3d  from  the  2d,      2y+2z=18 
From  these  two,  to  derive  one, 

6.  Subtract  the  5th  from  the  4th,  z=5. 

To  find  x  and  y,  we  have  only  to  take  their  values  from 
the  third  and  fifth  equations.     (Art.  206.) 

7.  Reducing  the  5th,  y=  9-z=9  —  5=4. 

8.  Transposing  in  the  3d,  #=12-z-2/=12-5— 4=3. 


CONTAINING    THREE    UNKNOWN    aUANTITIES.    117 

Prob.  27.  To  find  x,  y  and  z,  from 

1.  The  equation  x+  y+  2  =  12  } 

2.  And  x+2y+3z=20  > 

3.  And  4^+iy-f  z=6    ) 

4.  Multiplying  the  1st  by  3,  3x+3y+3z=36 

5.  Subtracting  the  2d  from  the  4th,  2x+  y=16 

6.  Subtracting  the  3d  from  the  1st,  x—\x+y—\y=6 

7.  Clearing  the  6th  of  fractions,  4a;+3y=36  ) 

8.  Multiplying  the  5th  by  3,  Gx+3y=48  ) 

9.  Subtracting  the  7th  from  the  8th,  2x=12.     And  x=6. 

^    ,      .  .  36-4x       36-24 

10.  Reducing  the  7th,  y=  — - —  =  — - — =4. 

o  o 

11.  Reducing  the  1st,  z=12-x-3/=12-6-4=2. 

In  this  example,  all  the  reductions  have  been  made  accor- 
ding to  the  third  rule  for  eliminating  unknown  quantities. 
(Art.  205.)    But  either  of  the  three  may  be  used  at  pleasure. 

SO  8.  A  calculation  may  often  be  very  much  abridged,  by 
the  exercise  of  judgment  in  stating  the  question,  in  selecting 
the  equations  from  which  others  are  to  be  deduced,  in  simpli- 
fying fractional  expressions,  &c.  The  skill  which  is  neces- 
sary for  this  purpose,  however,  is  to  be  acquired,  not  from  a 
system  of  rules,  but  from  practice,  and  a  habit  of  attention 
to  the  peculiarities  in  the  conditions  of  different  problems, 
the  variety  of  ways  in  which  the  same  quantity  may  be  ex- 
pressed, the  numerous  forms  which  equations  may  assume, 
&c.  In  many  of  the  examples  in  this  and  the  preceding  sec- 
tion, the  processes  might  have  been  shortened.  But  the  ob- 
ject has  been  to  illustrate  general  principles  rather  than  to 
furnish  specimens  of  expeditious  solutions.  The  learner  will 
do  well,  as  he  passes  along,  to  exercise  his  skill  in  abridging 
the  calculations  which  are  given,  or  substituting  others  in 
their  stead. 

(x+y— z=<z*\ 

Prob.  28.     Given      lx+z— y=b\  To  find  x,  y  and  z. 

(jy+z— x=c) 

.  a-\-b  a+c  b+c 

Ans.  *=__,  y=_,  %=__. 


118 


SIMPLE      EdUATIONS 


Prob.  29.  To  divide  46  into  three  such  parts,  that  if  10  be 
added  to  the  first  and  second,  they  shall  be  in  the  ratio  of  5 
to  9,  and  if  3  be  taken  from  the  second  and  third,  they  shall 
be  in  the  ratio  of  2  to  3.  Ans.  5,  17  and  24. 

209.  The  learner  must  exercise  his  own  judgment,  as  to 
the  choice  of  the  quantity  to  be  first  eliminated.  It  will  gen- 
erally be  best  to  begin  with  that  which  is  most  free  from  co- 
efficients, fractions,  &c. 

Prob.  30.  A  says  to  B,  give  me  10  dollars  and  I  shall  have 
twice  as  much  money  as  you;  B  says  to  C,  give  me  20  dol- 
lars and  I  shall  have  thrice  as  much  as  you ;  C  says  to  A, 
give  me  6  dollars,  and  I  shall  have  five  times  as  much  as 
you.     How  much  has  each  ? 

Ans.  A  $14,  B  $22,  C  $34. 


Prob.  31.  To  find  x,  y  and  %,  from 

The  equation  #+iy+iz=18' 

And  y+^+i^=23 

And  2+is+iy=25, 

Ans.  x=6,  y=12,  z=18. 


Prob.  32.  Given     «< 


]       1        "\ 

-  +  -=a 
x      y 

-  +  -=b  V  To  find  x,  y  and  z. 


1       1 

,  -  +  -=c 


Ans.  x- 


a-\-b—c 


~a-\-c—b 


%— 


b+c—a 


Prob.  33.  In  a  number  consisting  of  three  digits,  the  mid- 
dle digit  is  half  the  sum  of  the  other  two.  If  the  number  be 
divided  by  the  sum  of  its  digits,  the  quotient  will  be  26 ;  and 
if  198  be  added  to  the  number,  the  order  of  the  digits  will 
be  reversed.     What  is  the  number  ?  Ans.  234. 

210.  The  same  method  which  is  employed  for  the  re- 
duction of  three  equations,  may  be  extended  to  four,  five,  or 
any  number  of  equations,  containing  as  many  unknown 
quantities. 


CONTAINING     SEVERAL    UNKNOWN    Q  U  A  N T I TI ES .    1 19 

The  unknown  quantities  may  be  eliminated,  one  after  an- 
other, and  the  number  of  equations  may  be  reduced  by  suc- 
cessive steps  from  five  to  four,  from  four  to  three,  from  three 
to  two,  &c. 

The  general  rule  for  the  reduction  of  n  equations  contain- 
ing n  unknown  quantities  may  be  stated  thus. 

Combine  any  one  of  the  equations  with  each  of  the  others, 
so  as  to  eliminate  in  each  case  the  same  unknown  quantity. 
There  will  then  be  n—l  new  equations,  containing  n—  1  un- 
known quantities. 

Eliminate  another  unknown  quantity  by  combining  one  of 
these  new  equations  with  each  of  the  others.  This  will  give 
7i—2  equations,  containing  n—2  unknown  quantities. 

Continue  this  process  till  there  is  obtained  a  single  equa- 
tion containing  one  unknown  quantity.  From  this  equation 
deduce  the  value  of  this  unknown  quantity ;  and  then  by 
going  back  to  preceding  equations,  determine  successively  the 
values  of  the  other  unknown  quantities. 

Prob.  34.  To  find  w,  x,  y  and  z  from 

1.  The  equation  x+2y—  z+  w=6 

2.  And  x—  y+3z  —  w=4 

3.  And  2w+  y-r2z—  x  =  15 

4.  And  x-h  y+  %+2w=14> 

Combining  the  first  equation  with  each  of  the  other  three, 

5.  3y—4z+2w=2 

6.  3y+  z+3w=2l 

7.  y—2z—   w==  —  S 
Eliminating  y  from  these  new  equations, 

8.  v  5%+  w=19 

9.  2z+5iv=26 
Hence                                        23z=69,  and  %=3. 

By  going  back  to  the  8th  or  9th  equation,  and  substituting 
for  z  its  value,  we  find  w=4.  The  value  of  y  is  next  ob- 
tained from  one  of  the  equations  5,  6,  and  7 ;  and  finally  the 
value  of  x  is  deduced  from  one  of  the  original  equations. 
These  values  are  2  and  1. 


120  SIMPLE      ECtUATIONS 

2 1 1 .  The  method  of  combining  the  equations  may  be 
somewhat  varied  to  suit  different  cases.  Instead  of  combin- 
ing one  equation  with  several  others,  as  the  rule  directs,  we 
may  arrange  the  equations  in  any  order,  and  combine  the 
second  with  the  first,  the  third  with  either  of  the  two  first, 
the  fourth  with  either  of  the  three  first,  and  so  on  to  the  last. 

Prob.  35.  To  find  u,  x,  y  and  z,  from  the  equations 

1.  9m— 4#-f-10y-}-3z=9 

2.  3z+6y—  2z+9u=6 

3.  4y+5z—  6m— 5#=5 

4.  2z— 3u—  3x+8y=3 

Ans.  u=%,  x=2,  y=i,  z=3. 

Prob.  36.  Divide  18  into  four  such  parts,  that  the  first 
with  half  the  sum  of  the  other  three  shall  be  10,  the  second 
with  a  third  of  the  sum  of  the  other  three  shall  be  8,  and  tKe 
third  with  twice  the  sum  of  the  other  three  shall  be  31. 

Ans.  The  parts  are  2,  3,  5  and  8. 

Prob.  37.  A  number  consists  of  four  digits  whose  sum  is 
19.  The  first  or  left-hand  digit  is  equal  to  the  sum  of  the 
second  and  third  ;  the  second  is  equal  to  the  sum  of  the  third 
and  fourth ;  and  if  8082  be  taken  from  the  number,  the  or- 
der of  the  digits  will  be  inverted.     What  is  the  number  ? 

Ans.  9541. 

212.  If  in  the  algebraic  statement  of  the  conditions  of 
a  problem,  the  original  equations  are  more  numerous  than 
the  unknown  quantities ;  these  equations  will  either  be  con- 
tradictory, or  one  or  more  of  them  will  be  superfluous. 

Thus  the  equations    j       ~       >    are  contradictory. 

For  by  the  first  #=20,  while  by  the  second  #=40. 

But  if  the  latter  be  altered,  so  as  to  give  to  x  the  same 
value  as  the  former,  it  will  be  useless,  •in  the  statement  of  a 
problem.  For  nothing  can  be  determined  from  the  one, 
which  cannot  be  from  the  other. 

Thus  of  the  equations    j  i^Ziq  (    one  is  superfluous. 


I 


I 


CONTAINING    SEVERAL    UNKNOWN    QUANTITIES.    121 

For  either  of  them  is  sufficient  to  determine  the  ^lue  of  x. 
They  are  not  independent  equations.  (Art.  201.)  One  is 
convertible  into  the  other.  For  if  we  divide  the  1st  by  6,  it 
will  become  the  same  as  the  second.  Or  if  we  multiply  the 
second  by  6,  it  will  become  the  same  as  the  first. 

213.  But  if  the  number  of  independent  equations  pro 
duced  from  the  conditions  of  a  question,  is  less  than  the  num- 
ber of  unknown  quantities,  the  question  is  not  sufficiently 
limited  to  admit  of  a  definite  answer.  For  each  equation 
can  limit  but  one  quantity.  And  to  enable  us  to  find  this 
quantity,  all  the  others  connected  with  it,  must  either  be  pre- 
viously known,  or  be  determined  from  other  equations.  If 
this  is  not  the  case,  there  will  be  a  variety  of  answers  which 
will  equally  satisfy  the  conditions  of  the  question.  If,  for 
instance,  x  and  y  are  required  from  the  equation 

#+y=I00, 

there  may  be  fifty  different  answers.  The  values  of  x  and  y 
may  be  either  99  and  1,  or  98  and  2,  or  97  and  3,  &c.  For  the 
sum  of  each  of  these  pairs  of  numbers  is  equal  to  100.  But 
if  there  is  a  second  equation  which  is  independent  of  the 
former,  as  x— t/=96,  then  x  and  y  are  determined  ;  their 
values  are  98  and  2.  No  other  values  will  satisfy  both 
equations. 

214.  For  the  sake  of  abridging  the  solution  of  a  prob- 
lem, however,  the  number  of  independent  equations  actually 
put  upon  paper  is  frequently  less    than  the  number  of  un 
known  quantities. 

Suppose  we  are  required  to  divide  100  into  two  such  parts, 
that  the  greater  shall  be  equal  to  three  times  the  less.  If 
we  put  x  for  the  greater,  the  less  will  be  100— x.    (Art.  199.) 

Then  by  the  supposition,  #=300— Sx. 

Tranposing  and  dividing,  a; =75,  the  greater. 

And  100—75=25,  the  less. 

Here,  two  unknown  quantities  are  found,  although  there 
appears  to  be  but  one  independent  equation.  The  reason  of 
this  is,  that  a  part  of  the  solution  has  been  omitted,  because 
it  is  so  simple,  as  to  be  easily  supplied  by  the  mind.  To 
have  a  view  of  the  whole,  without  abridging,  let  x—  the 
greater  number,  and  y=  the  less. 

11 


122  SIMPLE      EdUATIONS 

1.  Thgp  by  the  supposition,  x  +y=  100  ) 

2.  And  Sy=x       5 

3.  Transposing  x  in  the  1st,  t/=100— a; 

4.  Dividing  the  2d  by  3,  V=\^ 

5.  Making  the  3d  and  4th  equal,     ^=100  — x 

6.  Multiplying  by  3,  #=300— Sx 

7.  Transposing  and  dividing,  #=75,  the  greater. 

8.  By  the  3d  step,  y=  100  -x=25,  the  less. 

By  comparing  these  two  solutions  with  each  other,  it  will 
be  seen  that  the  first  begins  at  the  6th  step  of  the  latter,  all 
the  preceding  parts  being  omitted,  because  they  are  too  sim- 
ple to  require  the  formality  of  writing  down. 

2 1  *>.  In  most  cases  also,  the  solution  of  a  problem  which 
contains  many  unknown  quantities,  may  be  abridged,  by  par- 
ticular artifices  in  substituting  a  single  letter  for  several. 

Suppose  four  numbers,  u,  x,  y  and  z,  are  required,  of  which 
The  sum  of  the  three  first  is  13 

The  sum  of  the  two  first  and  last  17 

The  sum  of  the  first  and  two  last  18 

The  sum  of  the  three  last  -     21 

Then     1.  u+x+y=13 

2.  u+x+z  =  l7 

3.  u+y+z  =  18 

4.  x+y+z=2l. 

Let  s  be  substituted  for  the  sum  of  the  four  numbers,  that  is, 
for  u+x+y+z.     It  will  be  seen  that  of  these  four  equations, 

The  first  contains  all  the  letters  except  z,  that  is,  5—2=13 
The  second  contains  all  except  y,  that  is,  s—y  =17 

The  third  contains  all  except  x,  that  is,  s—x=18 

The  fourth  contains  all  except  u,  that  is,  5—^=21. 

Adding  all  these  equations  together,  we  have 
45—  z—y—x  —  u  =69 
Or  4s-(«+y+a:+H)=69  (Art.  83.) 
But  s=(z+y+x+u)  by  substitution. 
Therefore  45—5=69;  that  is,  35=69,  and  5=23. 


CONTAINING    SEVERAL     UNKNOWN    ClUANTITIES.    123 

Then  putting  23  for  s,  in  the  four  equations  in  which  it  is 
first  introduced,  we  have 

23-2=13^1  fz=23- 13=10 


23"^17L  Therefore^  ^23-17=6 
23-z=18  |  ^  x=23-18=5 

23-^=21  J  ^=23-21=2. 

Contrivances  of  this  sort  for  facilitating  the  solution  of 
particular  problems,  must  be  left  to  be  furnished  for  the  occa- 
sion, by  the  ingenuity  of  the  learner.  They  are  of  a  nature 
not  to  be  taught  by  a  system  of  rules. 

216.  In  the  resolution  of  equations  containing  several 
unknown  quantities,  there  will  often  be  an  advantage  in 
adopting  the  following  method  of  notation. 

The  co-efficients  of  one  of  the  unknown  quantities  are 
represented, 

In  the  first  equation,  by  a  single  letter,  as  a, 

In  the  second,  by  the  same  letter  marked  with  an  accent,  as  a', 

In  the  third,  by  the  same  letter  with  a  double  accent,  as  a",  &c.  | 

The  accented  letters  are  called  a  prime,  a  second,  a 
third,  &c. 

The  co-efficients  of  the  other  unknown  quantities,  are 
represented  by  other  letters  marked  in  a  similar  manner ;  as 
are  also  the  terms  which  consist  of  known  quantities  only. 

Two  equations  containing  the  two  unknown  quantities  x 
and  y  may  be  written  thus, 

ax  +  by=c 
a'x+b'y=cl 
Three  equations  containing  x,  y  and  z,  thus, 

ax-{-by-\-cz=d  i 

a'x-\-b'yJrc'z=d' 
a"x+b"y+c"z=d" 
Four  equations  containing  x,  y,  z  and  u,  thus, 
ax+by-{-cz-\-du=e 
a'x + b'y + c'z + d'u  ==  e' 
a"x+b"y+c"z+d"u=e'' 
a,"x+b'"y+c",z+d'"u=ef". 


121  SIMPLE      EQUATIONS 

The  same  letter  is  made  the  co-efficient  of  the  same  un- 
known quantity,  in  different  equations,  that  the  co-efficients 
of  the  several  unknown  quantities  may  be  distinguished,  in 
any  part  of  the  calculation.  But  the  letter  is  marked  with 
different  accents,  because  it  actually  stands  for  different 
quantities. 

Thus  we  may  put  a=4,  a'=6,  a"  =  10,  a'"=20,  &c. 
To  find  the  value  of  x  and  y 

1.  In  the  equation,  ax  +  by  =c    ) 

2.  And  m  a'x+  b'y=c'  ) 

3.  Multiplying  the  1st  by  &',  (Art.  205.)  ab'x+bb'y=cb' 

4.  Multiplying  the  2d  by  b,  ba'x+bbfy—bcr 

5.  Subtracting  the  4th  from  the  3d,         ab'x—ba'x—cV—bc1 

cb'—bc'^\ 

6.  Dividing  by  ab'—ba\  (Art.  124.)     x=   ,,     ,   , 

_         .    ._  ac'—cd 

By  a  similar  process,  V~~h'—\ 

The  symmetry  of  these  expressions  is  well  calculated  to 
fix  them  in  the  memory.  The  denominators  are  the  same  in 
both ;  and  the  numerators  are  like  the  denominators,  except 
a  change  of  one  of  the  letters  in  each  term. 

But  the  particular  advantage  of  this  method  is,  that  the 
expressions  here  obtained  may  be  considered  as  general  so- 
lutions, which  give  the  values  of  the  unknown  quantities,  in 
other  equations,  of  a  similar  nature. 

Thus  if  10x+6y=100 

And  40z+4y=200 

Then  putting     a=10  b  =6  c=100 

a'=40  b'=4  c'=200 

-Tr    -  cb' -be'       100X4-6X200 

We  have     x=—r, — r~,  =  ~rr — : — ^ — 7^-=4- 
ab'  —  ba'         10X4—6X40 

.     ,  ac'-ca'       10X200-100X40      , 

And  y=ab^ba->  =  -10X1^6X40"  =1°* 


CONTAINING    SEVERAL    UNKNOWN    QUANTITIES.    125 


Demonstration   of  Theorems. 

217.  Equations  have  been  applied,  in  this  and  the  pre- 
ceding sections,  to  the  solution  of  problems.  They  may  be 
employed  with  equal  advantage,  in  the  demonstration  of  theo- 
rems. The  principal  difference,  in  the  two  cases,  is  in  the 
order  in  which  the  steps  are  arranged. 

In  solving  a  problem,  the  object  is  to  find  the  value  of  the 
unknown  quantity,  by  disengaging  it  from  all  other  quanti- 
ties. But,  in  conducting  a  demonstration,  it  is  necessary  to 
bring  the  equation  to  that  particular  form  which  will  express, 
in  algebraic  terms,  the  proposition  to  be  proved. 

Ex.  1.  Theorem.  Four  times  the  product  of  any  two  num- 
bers, is  equal  to  the  square  of  their  sum,  diminished  by  the 
square  of  their  difference. 

Let  x=  the  greater  number,  s=  their  sum, 

t/=  the  less,  d=  their  difference, 

p=  their  product. 

Demonstration. 

1.  }  Cx+y=s 

2.  >  By  the  notation  lx—y=d 

3.  j  (     xy=p 

4.  Adding  the  1st  and  2d  2x—s+d 

5.  Subtracting  the  2d  from  the  1st       2y=s—d 

6.  Multiplying  the  4th  and  5th  4xy=(s+d)  (s—d) 

7.  BysubstitutionandArt.lll    •        4p=s2—d2. 

The  last  equation  expressed  in  words  is  the  proposition  which 
was  to  be  demonstrated. 

It  will  be  seen  that  the  demonstration  consists  in  first  ex- 
pressing p,  s,  and  d  in  terms  of  x  and  y  by  means  of  three 
equations,  and  then  eliminating  x  and  y,  so  as  to  obtain  an 
equation  in  terms  of  p,  s  and  d  only.  The  final  equation 
must  be  of  this  kind,  since  the  proposition  felates  not  to  the 
quantities  x  and  y,  but  only  to  their  product,  sum  and  dif- 
ference. 

11* 


126  SIMPLE      EaUATIONS 

The  theorem  is  applicable  to  any  two  numbers  whatever. 
For  the  particular  values  of  x  and  y  will  make  no  difference 
in  the  nature  of  the  proof. 

Thus         4X   8X   6=(  8+  6)2-(  8-  6)2  =  192. 
And  4X10X   6=(10  +  6)2-(10-   6)2=240. 

And  4X12X10=(12  +  10)2-(12~10)2=480. 

Theorem  2.  The  sum  of  the  squares  of  two  numbers  is 
equal  to  the  square  of  their  sum,  diminished  by  twice  their 
product. 

318.  General  propositions  are  also  discovered,  in  an  ex- 
peditious manner,  by  means  of  equations.  The  relations  of 
quantities  may  be  presented  to  our  view,  in  a  great  variety 
of  ways,  by  the  several  changes  through  which  a  given  equa- 
tion may  be  made  to  pass.  Each  step  in  the  process  will 
contain  a  distinct  proposition. 

Let  s  and  d  be  the  sum  and  difference  of  two  quantities 
x  and  y. 

1.  Then  s=x+y 

2.  And  d=x—y 

3.  Dividing  the  first  by  2,  ^s=^x+%y 

4.  Dividing  the  2d  by  2,  id=^x—^y 

5.  Adding  the  3d  and  4th,  \s+\d=\x+\x=x 

6.  Subtracting  the  4th  from  the  3d,  \s— id=^y+^y=y. 

That  is, 

Half  the  sum  of  two  quantities  increased  by  half  their 
difference  is  equal  to  the  greater  ;  and 

Half  their  sum  diminished  by  half  their  difference  is  equal 
to  the  less. 


INVOLUTION      AND      EVOLUTION.  127 


SECTION    IX. 


INVOLUTION    AND    EVOLUTION. 

Art.  21 9.  When  a  quantity  is  multiplied  into  itself, 
the  product  is  called  a  power. 

For  the  notation  of  powers  see  Arts.  36,  7. 

The  scheme  of  notation  by  exponents  has  the  peculiar 
advantage  of  enabling  us  to  express  an  unknown  power. 
For  this  purpose  the  index  is  a  letter,  instead  of  a  numerical 
figure.  In  the  solution  of  a  problem,  a  quantity  may  occur, 
which  we  know  to  be  some  power  of  another  quantity.  But 
it  may  not  be  yet  ascertained  whether  it  is  a  square,  a  cube, 
or  some  higher  power.  Thus  in  the  expression  az,  the  index 
x  denotes  that  a  is  involved  to  some  power,  though  it  does  not 
determine  what  power.  So  bm  and  dn  are  powers  of  b  and  d; 
and  are  read  the  mth  power  of  b,  and  the  nth  power  of  d. 
When  the  value  of  the  index  is  found,  a  number  is  generally 
substituted  for  the  letter.  Thus  if  m=3  then  bm  =  b3 ;  but 
if  m=5,  then  bm=b5. 

320.  The  method  of  expressing  powers  by  exponents  is 
also  of  great  advantage  in  the  case  of  compound  quantities. 
Thus  a+b+d\3  or  a+b+d3  or  (a+b+d)3,  is  (a+b+d)X 
(a+b+d)  X  (a+b+d)  that  is,  the  cube  of  (a+b+d).  But 
this  involved  at  length  would  be 

a3+3a2b  +  3a2d+3ab2+6abd+3ad2+b3+3b2d+3bd2+d3. 

221.  If  we  take  a  series*  of  powers  whose  indices  in- 
crease or  decrease  by  1,  we  shall  find  that  the  powers  them- 
selves increase  by  a  common  multiplier,  or  decrease  by  a 
common  divisor;  and  that  this  multiplier  or  divisor  is  the 
original  quantity  from  which  the  powers  are  raised. 

Thus  in  the  series  aaaaa,     aaaaf     aaay     aa,     1 ; 

Or  a5  a*        a3      a2    a1; 

*  Note. — The  term  series  is  applied  to  a  number  of  quantities  succeeding 
each  other,  in  some  regular  order.  It  is  not  confined  to  any  particular  law  of 
increase  or  decrease. 


128        i  INVOLUTION      AND      EVOLUTION. 

the  indices  counted  from  right  to  left  are  1,  2,  3,  4,  5 ;  and 
the  common  difference  between  them  is  a  unit.  If  we  begin 
on  the  right  and  multiply  by  a,  we  produce  the  several 
powers,  in  succession,  from  right  to  left. 

Thus  a  Xa=a2  the  second  term.     And  a3Xa=a* 

a2Xa~a3  the  third  term.  a*Xa=a5,  &c. 

If  we  begin  on  the  left,  and  divide  by  a, 

We  have  as-—a=a4  And  a3-±-a=a2 

a*-±-a=a3  a2-r-a=al 

222*  But  this  division  may  be  carried  still  farther ;  and 
we  shall  then  obtain  a  new  set  of  quantities. 

Thus  a+a=-  =  1.     (Art.  123.)    ~-r-a= —     (Art.  166.) 

a  v  '    a  aa      v  ' 

1  lie 

\-r-a--  ra= ,  &c. 

a  aa  aaa 

The  whole  series  then 

1      l         1  1  M 

is    aaaaa,  aaaa,  aaa,  aa,  a,  1,  ->    — >    >    &c. 

a     aa     aaa 

111c 

Or  a5,  a*,  a3,  a2,  a,  1,  ->   —  >   —  %    &c. 

'       '  a     a2     a3 

Here  the  quantities  on  the  right  of  1,  are  the  reciprocals 
of  those  on  the  left.  (Art.  43.)  The  former,  therefore,  may 
be  properly  called  reciprocal  powers  of  a ;  while  the  latter 
may  be  termed,  for  distinction's  sake,  direct  powers  of  a.  It 
may  be  added,  that  the  powers  on  the  left  are  also  the  recip- 
rocals of  those  on  the  right. 

For  l-i--=lX?=a.     (Art.  165.)    And  lH- —  =as. 
a  1  x  a3 

la2  1 

a2  1  a4 

223.  The  same  plan  of  notation  is  applicable  to  com- 
pound quantities.     Thus  from  a-\-b,  we  have  the  series, 

{a+b)3>  {a+by,  (8+6)i  i,  _^T),  _*_,  -^-L-,  &c. 


INVOLUTION      AND      EVOLUTION.  129 

221.  For  the  convenience  of  calculation,  another  form 
of  notation  is  given  to  reciprocal  powers. 

According  to  this,  -  or  — -=a~1.     And  or  -r=ar3. 

a       a1  aaa       a3 

—  or  — =cr2.  or  —-=a~4,  &c. 

aa       a2  aaaa       a* 

And  to  make  the  indices  a  complete  series,  with  1  for  the 

common  difference,  the  term  -  or  1,  which  is  considered  as 

a 

no  power,  is  written  a0.  Though  a°  has  no  more  effect  as 
a  factor  than  unity,  yet  it  is  sometimes  expedient  to  retain 
it,  in  connection  with  other  letters ;  for  the  purpose  of  indi- 
cating, that  it  is  the  result  of  a  division,  where  equal  powers 
of  a  quantity  had  been  in  both  the  dividend  and  divisor. 

am  6b*c*d5 

Thus     —  =  am-m=a\  And     ^-7-^37  =3b°cd<>. 

am  2o*c2d5 

The  powers  both  direct  and  reciprocal*  then, 

t  2     c  a     1       1         1  1         # 

Instead  01   aaaa.  aaa.  aa*  a*    — »   —  >    — >    >   ,    <&c. 

a     a     aa     aaa     aaaa 

Will  be  a*,  a3,  a2,  a1,  a0,  a"1,  a"2,  a"3,  a~\  &c. 

Or  a+4,  a+3,  a+2t  a+l,  a\  a~\  a'2,  a"3,  a'\  &c. 

And  the  indices  taken  by  themselves  will  be, 

+4,  +3,  +2,  +1,  +0,  -1,  -2,  -3,  -4,  &c. 

225.  The  root  of  a  power  may  be  expressed  by  more 
letters  than  one. 

Thus    aaXaa,   or  aa\2   is  the  second  power  of  aa. 
And  aaXaaXaa,  or  aa\3  is  the  third  power  of  aa,  &c. 

Hence  a  certain  power  of  one  quantity,  may  be  a  different 
power  of  another  quantity.  Thus  a4  is  the  second  power 
of  a2,  and  the  fourth  power  of  a. 

22  G.  All  the  powers  of  1  are  the  same.  For  1X1,  or 
IX 1X1,  &c.  is  still  1. 

*  See  ISTote  E. 


130  INVOLUTION      AND      EVOLUTION. 

227.  Involution  is  finding  any  power  of  a  quantity,  by 
multiplying  it  into  itself.  The  reason  of  the  following  gene- 
ral rule  is  manifest,  from  the  nature  of  powers. 

Multiply  the  quantity  into  itself,  till  it  is  taken  as  a  factor, 
as  many  times  as  there  are  units  in  the  index  of  the  power 
to  which  the  quantity  is  to  be  raised. 

This  rule  comprehends  all  the  instances  which  can  occur 
in  involution.  But  it  will  be  proper  to  give  an  explanation 
of  the  manner  in  which  it  is  applied  to  particular  cases. 

228.  A  single  letter  is  involved,  by  giving  it  the  index 
of  the  proposed  power ;  or  by  repeating  it  as  many  times,  as 
there  are  units  in  that  index. 

The  4th  power  of  a,  is  a4  or  aaaa. 

The  6th  power  of  y,  is  yQ,  or  yyyyyy. 

The  7ith  power  of  x,  is  xn  or  xxx . . .  n  times  repeated. 

229.  The  method  of  involving  a  quantity  which  consists 
of  several  factors,  depends  on  the  principle,  that  the  power 
of  the  product  of  several  factors  is  equal  to  the  product  of 
their  powers. 

Thus  (ay)2=a2y2.     For  by  Art.  227  ;  (ay)2—ayXay. 
But  ayXay=ayay—aayy=a2y2. 

So      (bmx) 3 = bmx  X  bmx  X  bmx = bbbmmmxxx =b3m3x3. 
And    (ady)n=ady  X  ady  X  ady  . . .  n  times = andnyn. 

In  finding  the  power  of  a  product,  therefore,  we  may  either 
involve  the  whole  at  once ;  or  we  may  involve  each  of  the 
factors  separately,  and  then  multiply  their  several  powers 
into  each  other. 

Ex.  1.  The  4th  power  of  dhy,  is  (dhy)A,  or  d*hAy*. 

2.  The  3d  power  of  4b,  is  (4b)3,  or  43b3,  or  64b 3. 

3.  The  nth  power  of  6ad,  is  (6ad)n,  or  6nandn. 

4.  The  3d  power  of  SmX2y,  is  (SmX2y) 3,  or27m3XSy3. 

230.  A  compound  quantity  consisting  of  terms  connected 
by  +  and  —,  is  involved  by  an  actual  multiplication  of  its 
several  parts.     Thus, 


INVOLUTION      AND      EVOLUTION-  131 

(a  +  b)  l=a  +  b,  the  first  power. 
a  +  b 


a2+ab 
+  ab+b2 

(a-\-b)2  =a2  -\-2ab+b2 ,  the  second  power  of  (a+b). 
a  +b 


a3+2a2b  +  ab2 
+  a2b+2ab2+b3 


(a+b)3  =  a3+3a2b+3ab2+b3,  the  third  power,  &c. 

2.  The  square  of  a— b,  is  a2—  2ab+b2. 

3.  The  cube  of  a-\-\  is  a3+3a2 +3a+l. 

4.  The  square  of  a+b+h,  is  a2  +2ab+2ah+b2 +2bh+h2. 

5.  Required  the  cube  of  ^+2^+3. 

6.  Required  the  4th  power  of  6+2. 

7.  Required  the  5th  power  of  £  +  1. 

8.  Required  the  6th  power  of  1—b. 

For  the  method  of  finding  the  higher  powers  of  binomials, 
see  one  of  the  succeeding  sections. 

S3 1 .  For  many  purposes,  it  will  be  sufficient  to  express 
the  powers  of  compound  quantities  by  exponents,  without  an 
actual  multiplication. 

Thus  the  square  of  a+b,  is  a+b\2  or  (a+b)2.  (Art.  220.) 
The  nth  power  of  bc+8+x,  is  (bc+$+x)n. 

In  cases  of  this  kind,  the  vinculum  must  be  drawn  over  all 
the  terms  of  which  the  compound  quantity  consists. 

232.  But  if  the  root  consists  of  several  factors,  the  vin- 
culum which  is  used  in  expressing  the  power,  may  either 
extend  over  the  whole ;  or  may  be  applied  to  each  of  the 
factors  separately,  as  convenience  may  require. 


Thus  the  square  of  a+,bXc+d,  is  either 
(a+b)X(c+d)\2   or  'a^b\2x'cTd\2. 


132  INVOLUTION      AND      EVOLUTION. 

For  the  first  of  these  expressions  is  the  square  of  the  pro- 
duct of  the  two  factors,  and  the  last  is  the  product  of  their 
squares.     But  one  of  these  is  equal  to  the  other.     (Art.  229.) 

The  cube  of  aXb+d  is  (aXb+d)3  or  a3X(b-\-d)3. 

When  a  quantity  whose  power  has  been  expresssed  by  a 
vinculum  and  an  index,  is  afterwards  involved  by  an  actual 
multiplication  of  the  terms,  it  is  said  to  be  expanded. 

Thus  (a+b)2,  when  expanded,  becomes  a2 -\-2ab-\-b2 . 
And  (a+b+h)2,  becomes  a2 +2ab+2ah+b2  +2bh+h2. 

233.  With  respect  to  the  sign  which  is  to  be  prefixed  to 
quantities  involved,  it  is  important  to  observe,  that  when  the 
root  is  positive,  all  its  powers  are  positive  also;  but  when  the 
root  is  negative,  the  odd  powers  are  negative,  while  the  even 
powers  are  positive. 

For  the  proof  of  this,  see  Art.  103. 

The  2d  power  of  —a  is  -\~a2 
The  3d  power  is  —a3 

The  4th  power  is  +a4 

The  5th  power  is  —a5,  &c. 

Hence  any  odd  power  has  the  same  sign  as  its  root.  But 
an  even  power  is  positive,  whether  its  root  is  positive  or 
negative. 

Thus     +aX+a=a2 

And       —aX—a  —  a2. 

234.  A  quantity  which  is  already  a  power,  is  involved 
by  multiplying  its  index,  into  the  index  of  the  power  to  which 
it  is  to  be  raised. 

1.  The  3d  power  of  a2,  is  a2*3=a6. 

For  a2=aa:  and  the  cube  of  aa  is  aaXaaXaa=aaaaaa=a6 ; 
which  is  the  6th  power  of  a,  but  the  3d  power  of  a2. 

2.  The  4th  power  of  a 3b2,  is  a3X*b2X*=al2b*. 

3.  The  3d  power  of  4  a2x,  is  64  a6x3. 

4.  The  4th  power  of  2a3 X Sx2 d,  is  IGa12  X81x*d*. 

5.  The  5th  power  of  (a  +  b)2,  is  (a+b)1  °. 


INVOLUTION      AND      EVOLUTION.  133 

6.  The  nth  power  of  a3,  is  a3n. 

7.  The  nth  power  of  (z— y)m,  is  (x— y)mn. 

8.  a3+b3\2=a6+2a3b3+b6.     (Art.  109.) 

9.  a*Xb*\2=a*Xb*.  10.  (a3b2h*)3=a*b6h12. 

235.  The  rule  is  equally  applicable  to  powers  whose  ex- 
ponents are  negative. 

Ex.  1.  The  3d  power  of  a-2,  is  a~2X3=a-*. 

For  a-2  =  —  >    (Art.  224.)     And  the  3d  power  of  this  is 

J_    _L    JL        *        _L 

aa      aa      aa      aaaaaa      a6  ~~ 

a8 

2.  The  4th  power  of  a2 b~ 3  is  a8£r12,  or  j— ■• 

3.  The  cube  of  2xny-m,  is  8#3ni/-3m. 

4.  The  square  of  b3x~l,  is  b6x~2. 

5.  The  nth  power  of  x~m,  is  armw,  or  -^- 

S3 6.  It  must  be  observed  here,  as  in  Art.  233,  th  it  if  the 
sign  which  is  prefixed  to  the  power  be  — ,  it  must  be  changed 
to  +,  whenever  the  index  becomes  an  even  number. 

Ex.  1.  The  square  of  —  a3,  is  +«6.  For  the  square  of 
—  a3,  is  —  a3X—  a3,  which,  according  to  the  rules  for  the 
signs  in  multiplication,  is  +a6. 

2.  But  the  cube  of  -a3  is  -a9.  For -a3X-a3X-a3=-«9. 

3.  The  square  of  —  xn,  is  +x2n. 

4.  The  nth  power  of  —a3,  is  dba3n. 

Here  the  power  will  be  positive  or  negative,  according  as 
the  number  which  n  represents  is  even  or  odd. 

23 7.  A  fraction  is  involved  by  involving  both  the  nume- 
rator and  the  denominator. 

a       a2 
1.  The  square  of  7  is  yr-     For,  by  the  rule  for  the  mul- 
tiplication of  fractions,  (Art.  158,) 


a     a      aa      a2 
12 


bXb      bb      b2° 


13  i  INVOLUTION      AND      EVOLUTION. 

2.  The  2d,  3d,  and  nth  powers  of  ->  are  — ->    —  and  *—• 

r  a  a2     a3  a* 

_,  -   2zr2    .     Sx 3r6 

3.  The  cube  of  — — »  is  -z=—r- 

3y  27y3 

iC2  7* 

4.  The  nth  power  of  — -  >  is 


fly™*         anymn 

5.  The  square  of        ^j,      *  is        (g+1)i      ■ 

6.  The  cube  of  -^r>  is  -^r—      (Art.  235.) 

238.  Examples  of  binomials,  in  which  one  of  the  terms 
is  a  fraction. 

1.  Find  the  square  of  x+%,  and  x—\,  as  in  Art.  109. 


4a 

*-* 

x2+\x 

a:2+^+i 

2 

X2— £a; 

x2—x+\ 
4 

2.  The  square  of  «+->  is  «2+-^-  +  a 

6     .  •       &2 

3.  The  square  of  x+  ->   is  #2 +frc-f— -• 

4.  The  square  of  x >    is  x2 1 -■ 

n  m  mm2 


239.  It  has  been  shown,  (Art.  168,)  that  a  fractional 
co-efficient  may  be  transferred  from  the  numerator  to  the 
denominator  of  a  fraction,  or  from  the  denominator  to  the 
numerator.  By  recurring  to  the  scheme  of  notation  for  re- 
ciprocal powers,  (Art.  224,)  it  will  be  seen  that  any  factor 
may  also  be,  transferred,  if  the  sign  of  its  index  be  changed. 

ax~2 
1.  Thus,  in  the  fraction  >   we  may  transfer  x  from 

.         y  J 

the  numerator  to  the  denominator. 


INVOLUTION      AND      EVOLUTION.  1  .'i\~> 

ax'2      a  a       I         a 

^or  =  -xx'2  =  'X  —  =  — -• 

y      y  y    x~    y*2 

2.  In  the  fraction  j— ->  we  may  transfer  y  from  the  de- 
nominator to  the  numerator. 

^        a        a       1       a  ay'3 

For  br=~bxr=hxy'3=^b- 

da'*         d                                    b        by'* 
3      — 4#     =  _^ . 

x3        x3a*  ayn         a 

240.  In  the  same  manner,  we  may  transfer  a  factor 
which  has  a  positive  index  in  the  numerator,  or  a  negative 
index  in  the  denominator. 

1.  Thus    — r— =7 — --     For  x2  is  the  reciprocal  of  x'3, 
b       ox'3 

(Arts.  222,  224,)  that  is,  x3  =  — 

ty"2         6   * 

24:1.  Hence  the  denominator  of  any  fraction  may  be 
entirely  removed,  or  the  numerator  may  be  reduced  to  a  unit, 
without  altering  the  value  of  the  expression. 

_.  a         1 

1.  Thus     r=v: — r>    or  ab~l. 

b      ba'1 

x'2  1 

2.  t-i=  -7i-^'  or  &nar2. 

3-    ^  =  wv^'orcVa   *  • 

Addition  and  Subtraction  of  Powers. 

242.  It  is  obvious  that  powers  may  be  added,  like  other 
quantities,  by  writing  them  one  after  another  with  their  signs. 

Thus  the  s^m  of  a3  and  b2,  is  a3+b2. 

And  the  sum  of  a2—  bn  and  h5—d*,  is  a2—  bn+h5—  d^ 


Therefore, 

ax3 
b 

a 

bx'2 

ad2 

ay2 

xy'2 

xd'2 

ISC)  INVOLUTION   AND   EVOLUTION. 

.Q'13«  The  same  powers  of  the  same  letters  are  like  quan- 
tities ;  (Art.  39,)  and  their  co-efficients  may  be  added  or 
subtracted,  as  in  Arts.  67,  69. 

Thus  the  sum  of  2a2  and  3a2  is  5a3. 

It  is  as  evident  that  twice  the  square  of  a,  and  three  times 
the  square  of  a,  are  five  times  the  square  of  a,  as  that  twice 
a  and  three  times  a,  are  five  times  a. 

To         —3x«y5         3bm         3a*yn         —5a3h*         3(a+y)n 
Add       —2z6y5         6bm     —la*yn  6a3h*         4(a+y)n 

Sum      —  5x6y5  —4a*yn  7(a+y)w 

S44.  But  powers  of  different  letters  and  different  powers 
of  the  same  letter,  must  be  added  by  writing  them  down  with 
their  signs. 

The  sum  of  a2  and  a3  is  a2-\-a3. 

It  is  evident  that  the  square  of  a,  and  the  cube  of  a,  are 
neither  twice  the  square  of  a,  nor  twice  the  cube  of  a. 

The  sum  of  a3bn  and  3a5b6,  is  a3bn+3a5b6. 

f££5m  Subtraction  of  powers  is  to  be  performed  in  the 
same  manner  as  addition,  except  that  the  signs  of  the  subtra- 
hend are  to  be  changed  according  to  Art.  75. 

From  2a4  -3bn         3h2b6         a3bn         5(a-A)6 

Subtract      —6a4  4bn         4h2b6         a3bn         2(a—h)6 


Difference      8a4  —  h2b6  3(a-A)6 

Multiplication  of  Powers. 

£246.  Powers  may  be  multiplied,  like  other  quantities,  by 
writing  the  factors  one  after  another,  either  with,  or  without, 
the  sign  of  multiplication  between  them.     (Art.  89.) 

Thus  the  product  of  a3  into  b2,  is  a3b2,  or  aaabb. 

Multiply    x~3         h2b~n         3a6 y2       dh3x~n     a2b3y2 
Into  am  a4  —  2x  4by*         a3b2y 


Product     amx~3  —Ga6xy2  a2b3y2a3b2y 


INVOLUTION      AND      EVOLUTION.  1 37 

The  product  in  the  last  example,   may  be  abridged,  by 
bringing  together  the  letters  which  are  repeated. 
It  will  then  become  a5b5y3. 

The  reason  of  this  will  be  evident,  by  recurring  to  the 
series  of  powers  in  Art.  224,  viz. 

a+4,  a+3,  a+2,  a+\  a\  a~\  a~2,  a'3,  a~A,  &c. 

Or,  which  is  the  same, 

i     l       l          1           l         x 
aaaa,  aaa.  aa,  a,   1,  — >    — >     >    >    <fcc. 

a     aa      aaa     aaaa 

By  comparing  the  several  terms  with  each  other,  it  will  be 
seen  that  if  any  two  or  more  of  them  be  multiplied  together, 
their  product  will  be  a  power  whose  exponent  is  the  sum  of 
the  exponents  of  the  factors. 

Thus  a2Xa3=aaXaaa=aaaaa=a5.  So  anXam=an+m. 
Hence, 

24:7.  Powers  of  the  same  root  may  be  multiplied,  by  add- 
ing  their  exponents. 

Thus  a2XaG=a2+6=a*.    And  x3Xx2  Xx=x3+2  +  l=x*. 
Mult.  x3-\-x2yJrxy2-\-y3  into  x— y.  Ans.  xA—y4. 

Mult.  4x2y  +  3xy—  1  into  2x2—x. 
Mult.  x3-\-x— 5  into  2x2+x+\. 

The  rule  is  equally  applicable  to  powers  whose  exponents 
are  negative. 

1.  Thus  a-2Xa'3=a'5.     That  is    —  X = ■ 

aa     aaa      aaaaa 

2.  r»xrm=rw~m.    That  is  -x-==--5. 

3.  -a-2Xa-3  =  —  a~5.  4.  a"2 Xa3=a3~2=al. 
5.  a~nXam=am-n.  6.  y~2Xy2=y°  =  l. 

Division  of  Powers. 

948. -Powers  may  be  divided,  like  other  quantities,  by 
rejecting  from  the  dividend  a  factor  equal  to  the  divisor ;  or 
by  placing  the  divisor  under  the  dividend,  in  the  form  of  a 
fraction. 

12* 


133  INVOLUTION      AND      EVOLUTION. 

Thus  the  quotient  of  a3b2  divided  by  b2,  is  a3. 
Divide  9a3?/4     12b3xn       a2b+3a2y*       dx(a—h+y) 

By  -3a*  2b3  a2  (a-h+y)3 


Quotient  — 3y4  b+3y*  d 

a5 

The  quotient  of  a5  divided  by  a3,  is  —  •     But  this  is  equal 

to  a2.     For,  in  the  series 

a+4,  a+3,  a+2,  a+1,  a0,  a"1,  a"2,  a~3,  a"4,  &c. 
if  any  term  be  divided  by  another,  the  index  of  the  quotient 
will  be  equal  to  the  difference  between  the  index  of  the  divi- 
dend and  that  of  the  divisor. 

m.  ,     aaaaa  ,  am 

Thus    a5-T-a3  = =  a2.     And  ant-i-an= —=am-\ 

aaa  an 

Hence, 

249.  A  power  may  be  divided  by  another  power  of  the 
same  root,  by  subtracting  the  index  of  the  divisor  from  that 
of  the  dividend. 

Thus  y3+y2=y3~2=y\     That  is  ~=*y. 

And    an+l+a=an+l-l=an.     That  is   —=a\ 

a 

And    xn~xn=xn'n=x°  =  l.     That  is   —  =  1. 

xn 

Divide  y2m         b6         8aw+m         an+3         12(b+y)n 

By  ym  b3         4am  a2  3(b+y)3 


Quotient         yn  2an  4(b+y)n~3 

250*    The  rule  is  equally  applicable  to  powers  whose 
exponents  are  negative. 

1.  The  quotient  of  a'5  by  a"3,  is  a"*. 

m,  .  .    1     1     1    aaa        aaa         1 

That  is : = V = =  —  • 

aaaaa     aaa     aaaaa        1    aaaaa      aa 

1     1    x3  1 

2.  -ar'-f-*-3  =  -*-2.  That  is  ~  +  -±-  =  — =— • 

—  X5        X3       —Xs       —x2 


INVOLUTION      AND      EVOLUTION.  139 

1  h 

3.  h*+h~l=*h*+l=*k*.     That  is  A2-r  T=h2X T=h3. 

h  1 

4.  6an+2a-3=3an+3.  5.  ba3-±a=ba2. 
6.  b3  +  b5=b*-5=b-2.                    7.  a4-rfl7=fl-3. 

8.  (a3+3/3)m~(a3+y3)n=(«3+3/3)m""n. 

9.  (6+.r)n-r  (&+*)  =  (fc+z)*"1. 

The  multiplication  and  division  of  powers,  by  adding  and 
subtracting  their  indices,  should  be  made  very  familiar ;  as 
they  have  numerous  and  important  applications,  in  the  higher 
branches  of  algebra. 

Evolution. 

25 1 .  If  a  quantity  is  multiplied  into  itself,  the  product 
is  a  power.  On  the  contrary,  if  a  quantity  is  resolved  into 
any  number  of  equal  factors,  each  of  these  is  a  root  of  that 
quantity. 

Thus  b  is  the  root  of  bbb ;  because  bbb  may  be  resolved 
into  the  three  equal  factors,  b,  and  b,  and  b. 

In  subtraction,  a  quantity  is  resolved  into  two  parts. 
In  division,  a  quantity  is  resolved  into  two  factors. 
In  evolution,  a  quantity  is  resolved  into  equal  factors. 

252.  A  root  of  a  quantity,  then,  is  a  factor,  which  mul- 
tiplied into  itself  a  certain  number  of  times,  will  produce  that 
quantity. 

The  number  of  times  the  root  must  be  taken  as  a  factor, 
to  produce  the  given  quantity,  is  denoted  by  the  name  of 
the  root. 

Thus  2  is  the  4th  root  of  16;    because  2X2X2X2  =  16, 
where  two  is  taken  four  times  as  a  factor,  to  produce  16. 
So  a3  is  the  square  root  of  a6  ;  for  a3Xa3=a*. 
And  a2  is  the  cube  root  of  a6 ;  for  a2  Xa2Xa2=a6. 
And  a  is  the  6th  root  of  a6 ;  for  aXaXaXaXaXa=a6. 

Powers  and  roots  are  correlative  terms.  If  one  quantity 
is  a  power  of  another,  the  latter  is  a  root  of  the  former.  As 
b3  is  the  cube  of  b,  b  is  the  cube  root  of  b3. 


140  INVOLUTION      AND      EVOLUTION. 

253.  There  are  two  methods  in  use,  for  expressing  the 
roots  of  quantities ;  one  by  means  of  the  radical  sign  y,  and 
the  other  by  a  fractional  index.  The  latter  is  generally  to  be 
preferred;  but  the  former  has  its  uses  on  particular  occasions. 

When  a  root  is  expressed  by  the  radical  sign,  the  sign  is 
placed  over  the  given  quantity,  in  this  manner,   ya. 

Thus  ya  is  the  2d  or  square  root  of  a. 

ya  is  the  3d  or  cube  root. 

nya  is  the  rath  root. 
And    V«-fy  is  the  nth  root  of  a+y. 

254:.  The  figure  placed  over  the  radical  sign,  denotes  the 
number  of  factors  into  which  the  given  quantity  is  resolved ; 
in  other  words,  the  number  of  times  the  root  must  be  taken 
as  a  factor  to  produce  the  given  quantity. 

So  that  %/aX  ya=a. 

And        yaxyaxya=a. 

And        yaxya. . .  .n  times  =a. 

The  figure  for  the  square  root  is  commonly  omitted ;  ya 
being  put  for  ya.  Whenever,  therefore,  the  radical  sign  is 
used  without  a  figure,  the  square  root  is  to  be  understood. 

2t>«>.  When  a  figure  or  letter  is  prefixed  to  the  radical 
sign,  without  any  character  between  them,  the  two  quantities 
are  to  be  considered  as  multiplied  together. 

Thus  2ya,  is  2X  ya,  that  is,  2  multiplied  into  the  root 
of  «,  or,  which  is  the  same  thing,  twice  the  root  of  a. 

And  xyb,  is  xX  yb,  or  x  times  the  root  of  b. 

When  no  co-efficient  is  prefixed  to  the  radical  sign,  1  is 
always  to  be  understood ;  ya  being  the  same  as  1  ya,  that 
is,  once  the  root  of  a. 

250.  The  method  of  expressing  roots  by  radical  signs, 
has  no  very  apparent  connection  with  the  other  parts  of  the 
scheme  of  algebraic  notation.  But  the  plan  of  indicating 
them  by  fractional  indices,  is  derived  directly  from  the  mode 
of  expressing  powers  by  integral  indices.  To  explain  this, 
let  a6  be  a  given  quantity.  If  the  index  be  divided  into 
any  number  of  equal  parts,  each  of  these  will  be  the  index 
of  a  root  of  a6. 


INVOLUTION      AND      EVOLUTION.  141 

Thus  the  square  root  of  a8  is  a3.  For,  according  to  the 
definition,  (Art.  252,)  the  square  root  of  a6  is  a  factor,  which 
multiplied  into  itself  will  produce  a6.  But  a3Xa3=a6. 
(Art.  247.)  Therefore,  a3  is  the  square  root  of  a6.  The 
index, of  the  given  quantity  a6,  is  here  divided  into  the  two 
equal  parts,  3  and  3.  Of  course,  the  quantity  itself  is  re- 
solved into  the  two  equal  factors,  a3  and  a3. 

The  cube  root  of  a6  is  a2.     For  a2Xa2Xa2=a6. 

Here  the  index  is  divided  into  three  equal  parts,  and  the 
quantity  itself  resolved  into  three  equal  factors. 

The  square  root  of  a2  is  a1   or  a.     For  aXa—a2. 

By  extending  the  same  plan  of  notation,  fractional  indices 
are  obtained. 

Thus,  in  taking  the  square  root  of  a1   or  a,  the  index  1  is 

divided  into  two  equal  parts,  \  and  \ ;  and  the  root  is  a* 

On  the  same  principle, 

1 
The  cube  root  of  a,  is  a2=3ya. 

The  nth  root,  is  an  =  y<z,  &c. 


And  the  nth  root  of  a+x,  is  (a+x)n  =  Va+x. 

257,  In  all  these  cases,  the  denominator  of  the  fractional 
index,  expresses  the  number  of  factors  into  which  the  given 
quantity  is  resolved. 

ill                        ii 
So  that  a3Xa3Xa3=a.     And    anXan n  times  =  a. 

25  8.  It  follows  from  this  plan  of  notation,  that 

ill  j_i  l4-l 

G2Xa2=fl"2"+2.     For   a2-t-2_0i   or  a% 

1111  _Ll_J-l 

^X^Xfl^a^3^3^1,  &c. 

where  the  multiplication  is  performed  in  the  same  manner  as 
the  multiplication  of  powers,  that  is,  by  adding  the  indices. 

259.  Every  root  as  well  as  every  power  of  1  is  1.  (Art. 
226.)  For  a  root  is  a  factor,  wThich  multiplied  into  itself 
will  produce  the  given  quantity.  But  no  factor  except  1  can 
produce  1,  by  being  multiplied  into  itself. 

So  that  ln,  1,  v^lj  Vh  &c.  are  all  equal. 


1-12  INVOLUTION      AND      EVOLUTION. 

26®.  Negative  indices  are  used  in  the  notation  of  roots, 
as  well  as  of  powers.     See  Art.  224. 

Thus   \=a~%    \=a~%    -I=a"^. 
a1  a*  an 

Powers  of  Roots. 

26  1 .  It  has  been  shown  in  what  manner  any  power  or 
root  may  be  expressed  by  means  of  an  index.  The  index 
of  a  power  is  a  whole  number.  That  of  a  root  is  a  fraction 
whose  numerator  is  1.  There  is  also  another  class  of  quan- 
tities which  may  be  considered,  either  as  powers  of  roots,  or 
roots  of  powers. 

Suppose  a?  is  multiplied  into  itself,  so  as  to  be  repeated 
three  times  as  a  factor. 

1     I    1    !    1  3 

The  product  a^2^2"  or  a?  (Art.  258,)   is  evidently  the 

cube  of  a2,  that  is,  the  cube  of  the  square  root  of  a.  This 
fractional  index  denotes,  therefore,  a  power  of  a  root.  The 
denominator  expresses  the  root,  and  the  numerator  the  power. 
The  denominator  shows  into  how  many  equal  factors  or  roots 
the  given  quantity  is  resolved  ;  and  the  numerator  shows  how 
many  of  these  roots  are  to  be  multiplied  together. 

Thus  a3  is  the  4th  power  of  the  cube  root  of  a. 

The  denominator  shows  that  a  is  resolved  into  the  three 
ill 
factors  or  roots  «3,   and  a'3,   and  a3.     And  the  numerator 

shows  that  four  of  these  are  to  be  multiplied  together ;  which 

will  produce  the  fourth  power  of  a*  ;  that  is, 

lli-i-i 
a'sXa3Xa'3Xa2=a3. 

i 
262.   As  a2  is  a  power  of  a  root,  so  it  is  a  root  of  a 
power.     Let  a  be  raised  to  the  third  power  a3.     The  square 

root  of  this  is  a2.  For  the  root  of  a3  is  a  quantity  which 
multiplied  into  itself  will  produce  a3. 

2         111 

But  according  to  Art.  261,  a2=a2Xa2Xa2 ;  and  this  mul- 
tiplied into  itself,  is 

a2  Xa2  Xa2  Xa2  Xa2  Xa2  =a3 . 


INVOLUTION      AND      EVOLUTION.  143 

3 

Therefore  a?  is  the  square  root  of  the  cube  of  a. 

m 

In  the  same  manner,  it  may  be  shown  that  an  is  the  mih 
power  of  the  nth  root  of  a ;  or- the  rcth  root  of  the  rath  power : 
that  is,  a  root  of  a  power  is  equal  to  the  same  power  of  the 
same  root.  For  instance,  the  fourth  power  of  the  cube  root 
of  a,  is  the  same  as  the  cube  root  of  the  fourth  power  of  a. 

^£6«$.  Roots,  as  well  as  powers,  of  the  same  letter,  may 
be  multiplied  by  adding  their  exponents.  It  will  be  easy  to 
see,  that  the  same  principle  may  be  extended  to  powers  of 
roots,  when  the  exponents  have  a  common  denominator. 

Thus   aTXaT=a!  +  f=aT. 
For         a1=a1Xa1. 

z         1         l  1 

And        a1=a1Xa1Xa'1. 

Therefore  aT  X  «T = a1  X  a1  X  «T  X  aJ  X  «T = aT. 

S04.  The  value  of  a  quantity  is  not  altered,  by  applying 

to  it  a  fractional  index  whose  numerator  and  denominator 

are  equal. 

z.       i       * 
Thus   a=a*=a2=an.     For  the  denominator  shows  that 

a  is  resolved  into  a  certain  number  of  factors ;  and  the  nu- 

n 

merator  shows  that  all  these  factors  are  included  in  an. 

3.  1  1  X 

Thus  a3=a2Xa3Xa3,  which  is  equal  to  a. 

n        J.        J.        i. 
And     an  —anXanXan  ...  .n  times. 

On  the  other  hand,  when  the  numerator  of  a  fractional 
index  becomes  equal  to  the  denominator,  the  expression  may 
be  rendered  more  simple  by  rejecting  the  index. 

n 

Instead  of  an,  we  may  write  a. 

££6«>.  The  index  of  a  power  or  root  may  be  exchanged, 
for  any  other  index  of  the  same  value. 

2  1 

Instead  of  a3,  we  may  put  ad . 

For  in  the  latter  of  thes#  expressions,  a  is  supposed  to  be 
resolved  into  twice  as  many  factors  as  in  the  former;  and 
the  numerator  shows  that  twice  as  many  of  these  factors  are 


144  INVOLUTION      AND.    EVOLUTION 

to  be  multiplied  together.     So  that  the  whole  value  is  not 
altered. 

Thus  x2=x°=xvi  &c.  that  is,  the  square  of  the  cube  root 
is  the  same,  as  the  fourth  power  of  the  sixth  root,  the  sixth 
power  of  the  ninth  root,  &c. 

So  a2=a2=a3=a  *  .  For  the  value  of  each  of  these 
indices  is  2. 

266.  From  the  preceding  article,  it  will  be  easily  seen, 
that  a  fractional  index  may  be  expressed  in  decimals. 

1  _5_ 

1.  Thus  a2=a10,  or  a0'5;  that  is,  the  square  root  is 
equal  to  the  5th  power  of  the  tenth  root. 

2.  cfi=al  00,  or  a0'25  ;  that  is,  the  fourth  root  is  equal  to 
the  25th  power  of  the  100th  root. 

3.  aJ=a°*  5.  a*=a1'9 

4.  a^=a3S  6.  a^'=a2'Js. 

In  many  cases,  however,  the  decimal  can  be  only  an  ap- 
proximation to  the  true  index. 

Thus  cfi=a°'s  nearly.  a*== a0323*3  very  nearly. 

In  this  manner,  the  approximation  may  be  carried  to  any 
degree  of  exactness  which  is  required. 

Thus  a*=a1-**"*.  tt-V-=fli.s7i42# 

These  decimal  indices  form  a  very  important  class  of  num- 
bers, called  logarithms. 

£67.  It  is  frequently  convenient  to  vary  the  notation  of 
powers  of  roots,  by  making  use  of  a  vinculum,  or  the  radical 
sign  v/.  In  doing  this,  we  must  keep  in  mind,  that  the  power 
of  a  root  is  the  same  as  the  root  of  a  power ;  and  also,  that 
the  denominator  of  a  fractional  exponent  expresses  a  root, 
and  the  numerator  a  power. 

Instead,  therefore,  of  a3,  we  may  write  (a3)2,  or  {a2)3, 
or  ya2. 

The  first  of  these  three  form% denotes  the  square  of  the 
cube  root  of  a ;  and  each  of  the  two  last,  the  cube  root  of 
the  square  of  a. 


INVOLUTION      AND      EVOLUTION.  145 

m  jjto  i 

So       a"=a*     r=am|*  =  Vam. 


And    (bx)i  =(b*x3)i=Vb*x*. 


And    (a+y)5  =  (a+y)-i\5=V(a+y)*. 

26  8  •  Evolution  is  the  opposite  of  involution.  One  is 
finding  a  power  of  a  quantity,  by  multiplying  it  into  itself. 
The  other  is  finding  a  root,  by  resolving  a  quantity  into  equal 
factors.  A  quantity  is  resolved  into  any  number  of  equal 
factors,  by  dividing  its  index  into  as  many  equal  parts 
(Art.  256.) 

Evolution  may  be  performed,  then,  by  the  following  gen- 
eral rule ; 

Divide  the  index  of  the  quantity  by  the  number  expressing 
the  root  to  be  found. 

Or,  place  over  the  quantity  the  radical  sign  belonging  to 
the  required  root. 

1.  Thus  the  cube  root  of  a6,  is  a2.     For  a2Xa2Xa*=a*. 

Here  6,  the  index  of  the  given  quantity,  is  divided  by  3, 
the  number  expressing  the  cube  root. 

2.  The  cube  root  of  a  or  a1,  is  a5  or  ya. 
For  aSxcfiXcfi,  or  2JaXVaxya=a. 

3.  The  5th  root  of  ab,  is  (ab)J  or   Vab.     • 

4.  The  rath  root  of  a2,  is  a*  or  Va2. 

5.  The  7th  root  of  2d-x,  is  (2d-x)$  or   V2d-x. 

6.  The  5th  root  of  (a-x)*,  is  {a-x)^  or  V(a-x)\ 

7.  The  cube  root  of  a2,  is  a0. 

8.  The  4th  root  of  a"1,  is  <arT. 

9.  The  cube  root  of  a3,  is  a9. 

m 

10.  The  nth  root  of  xOT,  is  xn. 

269.  According  to  the  rule  just  given,  the  cube  root  of 
the  square  root  is  found,  by  dividing  the  index  £  by  3,  as  in 

13 


146  INVOLUTION      AND      EVOLUTION. 

example  7th.     But  instead  of  dividing  by  3,  we  may  multiply 
byf     For  i-3-i-f-iXf     (Art.  165.) 

So  — '-n=  ~  X  -•      Therefore  the  mth  root  of  the  nth 
m  m       n 

~  X  — 
root  of  a  is  equal  to  an     m. 

i 

Here  the  two  fractional  indices  are  reduced  to  one  by 
multiplication. 

It  is  sometimes  necessary  to  reverse  this  process ;  to  re- 
solve an  index  into  two  factors. 

Thus  x^=x^X^=x^\  .  That  is,  the  8th  root  of  x  is  equal 
to  the  square  root  of  the  4th  root. 


So  (a+b)™  =  (a+b)"X"=(a+b)"\*. 

It  may  be  necessary  to  observe,  that  resolving  the  index 
into  factors,  is  not  the  same  as  resolving  the  quantity  into 
factors.  The  latter  is  effected,  by  dividing  the  index  into 
parts. 

370.  The  rule  in  Art.  268,  may  be  applied  to  every  case 
in  evolution.  But  when  the  quantity  whose  root  is  to  be 
found,  is  composed  of  several  factors,  there  will  frequently 
be  an  advantage  in  taking  the  root  of  each  of  the  factors 
separately. 

This  is  done  upon  the  principle  that  the  root  of  the  product 
of  several  factors,  is  equal  to  the  product  of  their  roots. 

Thus  Vab=  y/aX  \/b.  For  each  member  of  the  equation 
if  involved,  will  give  the  same  power. 

The  square  of  ^ab,  is  ab.     (Art.  252.) 

The  square  of  </aX  Vb,  is   y/aX  \/aX  \/bX  \/b. 

But    y/aX\/a=a.     And   y/bx  y/b=b. 

Therefore  the  square  of  y/aX  y/b=y/aX  \/aX  y/bx  \/b 
=«6,  which  is  also  the  square  of  ^ ab. 

1  JL    J. 

On  the  same  principle,  (ab)n=anbn. 


INVOLUTION      AND      EVOLUTION.  147 

When,  therefore,  a  quantity  consists  of  several  factors,  we 
may  either  extract  the  root  of  the  whole  together ;  or  we 
may  find  the  root  of  the  factors  separately,  and  then  multiply 
them  into  each  other. 

Ex.  1.  The  cube  root  of  xy,  is  either  (xy)*  or  x*y*. 

2.  The  5th  root  of  3y,  is  V3y  or  V3X  %/y. 

3.  The  6th  root  of  abh,  is  (abh)*  or  d*b*h*. 

4.  The  cube  root  of  8b,  is  (8b) *  or  2b*. 

5.  The  ?ith  root  of  xny,  is  (xny)"  or  asy*. 

271.  The  root  of  a  fraction  is  equal  to  the  root  of  the 
numerator  divided  by  the  root  of  the  denominator. 

i  i        i    » 

mi        i  r  a      a2       _      a2      a2       a 

1 .  Thus  the  square  root  of  -  =  — .     For  —  X  —  =  t- 

b      b2  V*      b^      b 

2.  bo  the  rath  root  of  t  =  -t-     For  — rX— 7.  .n  times=T- 

b      £  6-    6"  6 

_.  1   x     ,      \/x  '1  /ah       ^  ah 

3.  1  he  square  root  of  — >  is  — =•       4.  v   — =    , - 

«y         Vay  *V      VXy 

272.  For  determining  what  sign  to  prefix  to  a  root,  it  is 
important  to  observe,  that 

An  odd  root  of  any  quantity  has  the  same  sign  as  the 
quantity  itself 

An  even  root  of  an  affirmative  quantity  is  ambiguous. 

An  even  root  of  a  negative  quantity  is  impossible. 

That  the  3d,  5th,  7th,  or  any  other  odd  root  of  a  quantity 
must  have  the  same  sign  as  the  quantity  itself,  is  evident 
from  Art.  233. 

373*  But  an  even  root  of  an  affirmative  quantity  may 
be  either  affirmative  or  negative.  For,  the  quantity  may  be 
produced  from  the  one,  as  well  as  from  the  other.   (Art.  233.) 

Thus  the  square  root  of  a2   is  +a  or  —a. 

An  even  root  of  an  affirmative  quantity  is,  therefore,  said 
to  be  ambiguous,  and  is  marked  with  both  +  and  — . 


148  INVOLUTION      AND      EVOLUTION. 

Thus  the  square  root  of  3b,  is  ±  ^Sb. 

1 
The  4th  root  of  x,  is  ±x^ 

The  ambiguity  does  not  exist,  however,  when,  from  the 
nature  of  the  case,  or  a  previous  multiplication,  it  is  known 
whether  the  power  has  actually  been  produced  from  a  posi- 
tive or  from  a  negative  quantity. 

274t.  But  no  even  root  of  a  negative  quantity  can  be 
found. 

The  square  root  of  —  a2  is  neither  -fa  nor  —a. 
For  +«X+«=+a2.     And  —  aX—a=+a2  also. 

» An  even  root  of  a  negative  quantity  is,  therefore,  said  to 
be  impossible  or  imaginary. 

There  are  purposes  to  be  answered,  however,  by  applying 
the  radical  sign  to  negative  quantities.  The  expression 
y/ — a  is  often  to  be  found  in  algebraic  processes.  For, 
although  we  are  unable  to  assign  it  a  rank,  among  either 
positive  or  negative  quantities  ;  yet  we  know  that  when  mul- 
tiplied into  itself,  its  product  is  —  a,  because  >/  —  a  is  by 
notation  a  root  of  —a,  that  is,  a  quantity  which  multiplied 
into  itself  produces  —a. 

This  may,  at  first  view,  seem  to  be  an  exception  to  the 
general  rule  that  the  product  of  two  negatives  is  affirmative. 
But  it  is  to  be  considered,  that  V  —  a  is  not  itself  a  negative 
quantity,  but  the  root  of  a  negative  quantity. 

The  mark  of  subtraction  here,  must  not  be  confounded 
with  that  which  is  prefixed  to  the  radical  sign.  The  expres- 
sion V— a  is  not  equivalent  to  — •  y/a.  The  former  is  a  root 
of  —  a ;  but  the  latter  is  a  root  of  +a : 

For  ~-  \/aX  —  \/a=y/aa=a. 
The  root  of  —  a,  however,  may  be  ambiguous.     It  may  be 
either  +\/  —  a,  or  —V  —  a. 

One  of  the  uses  of  imaginary  expressions  is  to  indicate  an 
impossible  or  absurd  supposition  in  the  statement  of  a  prob- 
lem. Suppose  it  be  required  to  divide  the  number  14  into 
two  such  parts,  that  their  product  shall  be  60.  If  one  of  the 
parts  be  x,  the  other  will  be  14— x.     And  by  the  supposition, 

xX(14— x)=60,  or  14x— x2=60. 


INVOLUTION      AND      EVOLUTION.  14$ 

This  reduced,  by  the  rules  in  the  following  section,  will  give 
#=7^^-11. 

As  the  value  of  x  is  here  found  to  contain  an  imaginary 
expression,  we  infer  that  there  is  an  iu consistency  in  the 
statement  of  the  problem  :  that  the  number  14  cannot  be 
divided  into  any  two  parts  whose  product  shall  be  00. 

27«>.  Mathematicians  are  not  entirely  agreed  in  the 
logical  explanations  of  imaginary  quantities.  It  appears  to 
be  taken  for  granted,  by  Euler  and  others,  that  the  product 
of  the  imaginary  roots  of  two  quantities  is  equal  to  the  root 
of  the  product  of  the  quantities;  for  instance,  that 

v/^ax  V^-b=  V—aX—b. 
If  this  principle  be   admitted,   certain  limitations    must  be 
observed  in  the  application.     If  we  make 

V  —  aX  y/  —  a=\/  —  aX—a, 
and  this,  in  conformity  with  the  common  rule  for  possible 
quantities,  =  \/a2  ;  yet  we  are  not  at  liberty  to  consider  the 
latter  expression  as  equivalent  to  a.  For  though  y/a2,  when 
taken  without  reference  to  its  origin,  is  ambiguous,  and  may 
be  either  -\-a  or  —a\  yet  when  we  know  that  it  has  been 
produced  by  multiplying  >/ —  a  into  itself,  we  are  not  permit- 
ted to  give  it  any  other  value  than  —a.     (Art.  273.) 

On  the  principle  here  stated,  imaginary  expressions  may  be 
easily  prepared  for  calculation,  by  resolving  the  quantity  under 
the  radical  sign  into  two  factors,  one  of  which  is  —  1 ;  thereby 
reducing  the  imaginary  part  of  the  expression  to  Yr-1.  As 
— .a=-\-aX  —  1,  the  expression  \/-fl=-/ax-l=\/aX^-l. 
So  V  —  a  —  b  —  >/a-\-bx  ^  — 1.  The  first  of  the  two  factors 
is  a  real  quantity.  After  the  impossible  part  of  imaginary 
expressions  is  thus  reduced  to  V— I,  they  may  be  multiplied 
and  divided  by  the  rules  already  given  for  other  radicals. 

Thus  in  Multiplication, 

1.  \/^aX  ^-1>=  y/aX  ^^Tx  VbX  vdr=r  %/abX-l  =  ->/ab. 

2.  -\-yS- ~aX  —  V~—b=—  y/abx  — 1  =  4V«&- 

4.   (l  +  v/3l)x(l-v/'==T)-2. 

13* 


150  INVOLUTION      AND      EVOLUTION. 

From  these  examples  it  will  be  seen,  that  according  to  the 
principle  assumed,  the  product  of  two  imaginary  expressions 
is  a  real  quantity. 

G.    v^2Xv/18=6X^:iT. 

Hence,  the  product  of  a  real  quantity  and  an  imaginary 
expression,  is  itself  imaginary. 

In  Division, 

1.  — =  =  - -=  =  */-.         2.    -t=  =  1. 

Hence,  the  quotient  of  one  imaginary  expression  divided 
by  another  is  a  real  quantity. 


\/—a       y/aX^  —  l 


/a        


</b  y/b 

y/a  s/a  1 

Hence,  the  quotient  of  an  imaginary  quantity  divided  by 
a  real  one,  or  of  a  real  quantity  divided  by  an  imaginary  one, 
is  itself  imaginary. 

By  multiplying  V  —  1  continually  into  itself,  we  obtain  the 
following  powers. 

&c.  &c. 

The  even  powers  being  alternately  —1  and  +1  and  the 
odd  powers,  —  V  —  l  and  +  >/  —  1. 

5£76.  The  methods  of  extracting  the  roots  of  compound 
quantities  are  to  be  considered  in  a  future  section.  But 
there  is  one  class  of  these,  the  squares  of  binomial  and  re- 
sidual quantities,  which  it  will  be  proper  to  attend  to  in  this 
place.     It  has  been  shown  (Art.  109,)  that  the  square  of  a 


INVOLUTION      AND      EVOLUTION.  151 

binomial  quantity  consists  of  three  terms,  two  of  which  are 
complete  powers,  and  the  other  is  a  double  product  of  the 
roots  of  these  powers.     The  square  of  a+b,  for  instance,  is 

a2+2ab+b2, 

two  terms  of  which,  a2  and  b2,  are  complete  powers,  and 
2ab  is  twice  the  product  of  a  into  b,  that  is,  the  root  of  aa 
into  the  root  of  b2. 

Whenever,  therefore,  we  meet  with  a  quantity  of  this  de- 
scription, we  may  know  that  its  square  root  is  a  binomial ; 
and  this  may  be  found,  by  taking  the  root  of  the  two  terms 
which  are  complete  powers,  and  connecting  them  by  the 
sign  +..  The  other  term  disappears  in  the  root.  Thus,  to 
find  the  square  root  of 


2 


x2+2xy+y 

take  the  root  of  x2,  and  the  root  of  y2,  and  connect  them 
by  the  sign  +.     The  binomial  root  will  then  be  x+y. 

In  a  residual  quantity,  the  double  product  has  the  sign  — 
prefixed,  instead  of  +.  The  square  of  a— b,  for  instance,  is 
a2  —2ab+b2.  (Art.  110.)  And  to  obtain  the  root  of  a  quantity 
of  this  description,  we  have  only  to  take  the  roots  of  the  two 
complete  powers,  and  connect  them  by  the  sign  — .  Thus 
the  square  root  of  x2—2xy+y2  is  x— y.     Hence, 

277.  To  extract  a  binomial  or  residual  square  root,  take 
the  roots  of  the  two  terms  which  are  complete  powers,  and 
connect  them  by  the  sign  which  is  prefixed  to  the  other  term. 

Ex.  1.  To  find  the  root  of  x2+2x+\. 

The  two  terms  which  are  complete  powers  are  x2   and  1. 
The  roots  are  x  and  1.     (Art.  259.) 
The  binomial  root  is,  therefore,  x+l. 

2.  The  square  root  of  x2  —  2x-\-\,  is  x—  1.     (Art.  110.) 

3.  The  square  root  of  a2+a+\,  is  a+\.     (Art.  238.) 

4.  The  square  root  of  a2+±a  +  ±,  is  tf-f§. 

b2  b 

5.  The  square  root  of  a2  +ab-\ >  is  a-\ 

4  2 

„    m,  r  2ab      b2     .  b 

6.  lhe  square  root  of  a2 -\ 1 >  is  a+-- 

c        c2  c 


152  INVOLUTION      AND      EVOLUTION. 

278.  A  root  whose  value  cannot  be  exactly  expressed  in 
numbers,  is  called  a  surd. 

Thus  v/2  is  a  surd,  because  the  square  root  of  2  cannot 
be  expressed  in  numbers,  with  perfect  exactness. 

In  decimals,  it  is  1.41421356  nearly. 

But  though  we  are  unable  to  assign  the  value  of  such  a 
quantity  when  taken  alone,  yet  by  multiplying  it  into  itself, 
or  by  combining  it  with  other  quantities,  we  may  produce 
expressions  whose  value  can  be  determined.  There  is,  there- 
fore, a  system  of  rules  generally  appropriated  to  surds.  But 
as  all  quantities  whatever,  when  under  the  same  radical  sign, 
or  having  the  same  index,  may  be  treated  in  nearly  the  same 
manner  ;  it  will  be  most  convenient  to  consider  them  together, 
under  the  general  name  of  Radical  Quantities  ;  understand- 
ing by  this  term,  every  quantity  which  is  found  under  a  radi- 
cal sign,  or  which  has  a  fractional  index. 

279.  Every  quantity  which  is  not  a  surd,  is  said  to  be 
rational.  But  for  the  purpose  of  distinguishing  between  radi- 
cals and  other  quantities,  the  term  rational  will  be  applied, 
in  this  section,  to  those  only  which  do  not  appear  under  a 
radical  sign,  and  which  have  not  a  fractional  index. 

Reduction  of  Radical  Quantities. 

280.  Before  entering  on  the  consideration  of  the  rules 
for  the  addition,  subtraction,  multiplication  and  division  of 
radical  quantities,  it  will  be  necessary  to  attend  to  the  meth- 
ods of  reducing  them  from  one  form  to  another. 

First,  to  reduce  a  rational  quantity  to  the  form  of  a  radical ; 

Raise  the  quantity  to  a  power  of  the  same  name  as  the  given 
root,  and  then  apply  the  corresponding  radical  sign  or  index. 

Ex.  1.  Reduce  a  to  the  form  of  the  nth  root. 

The  nth  power  of  a  is  an.     (Art.  228.) 

Over  this,  place  the  radical  sign,  and  it  becomes  yan. 

It  is  thus  reduced  to  the  form  of  a  radical  quantity,  with- 

n 

out  any  alteration  of  its  value.*  For  ny/an=an=a. 

2.  Reduce  4  to  the  form  of  the  cube  root.  , 

Ans.     ^64  or  (64) * 


INVOLUTION      AND      EVOLUTION.  153 

3.  Reduce  3a  to  the  form  of  the  4th  root.  

Ans.  Vsia4. 

4.  Reduce  \ab  to  the  form  of  the  square  root.  j 

Ans.  (\a2b2)2. 

5.  Reduce  SXa—x  to  the  form  of  the  cube  root. 

Ans.    \/27X(a-x)3.     See  Art.  229. 

6.  Reduce  a2  to  the  form  of  the  cube  root. 
The  cube  of  a2  is  a6.  • 
And  the  cube  root  of  a6  is   3/a*=a*\*. 

In  cases  of  this  kind,  where  a  power  is  to  be  reduced  to 
the  form  of  the  nth  root,  it  must  be  raised  to  the  nth  power, 
not  of  the  given  letter,  but  of  the  power  of  the  letter. 

Thus  in  the  example,  a6  is  the  cube,  not  of  a,  but  of  a2. 

7.  Reduce  a3bA  to  the  form  of  the  square  root. 

8.  Reduce  am  to  the  form  of  the  nth  root. 

381.  Secondly,  to  reduce  quantities  which  have  different 
indices,  to  others  of  the  same  value  having  a  common  index ; 

1.  Reduce  the  indices  to  a  common  denominator. 

2.  Involve  each  quantity  to  the  power  expressed  by  the 
numerator  of  its  reduced  index. 

3.  Take  the  root  denoted  by  the  common  denominator. 

Ex.  1.  Reduce    a*  and  b*  to  a  common  index. 

1st.  The  indices  £  and  }  reduced  to  a  common  denomina- 
tor, are  732  and  yV     (Art.  150.) 

2d.  TKe  quantities  a  and  b  involved  to  the  powers  ex- 
pressed by  the  two  numerators,  are  a3  and  b2 . 

3d.  The  root  denoted  by  the  common  denominator  is  T\. 

The  answer,  then,  is   a3|T2  and  62|T2~. 

The  two  quantities  are  thus  reduced  to  a  common  index, 
without  any  alteration  in  their  values. 


For  by  Art.  265,  a*=al  2,  which  by  Art.  269,  =aa 
And  universally  an  =  amn=am\mn. 

1  2. 

2.  Reduce    a2  and  (dx)2  to  a  common  index. 


15i  INVOLUTION      AND      EVOLUTION. 

The  indices  reduced  to  a  common  denominator  are  f  and  f. 

3  1  1  1 

The  quantities  then,  are  a6  and  (bx)6f  or  a2\\  and»&4x4|\ 

I  i  JL 

3.  Reduce  a2   and  6\  Ans.  a2n\n  and  6n. 

4.  Reduce  x±n  and  y~  Ans.  a^jm\    And  y*|*. 

5.  Reduce  3*  and  3*.  Ans.  8T  and  9°. 


6.  Reduce  (a+&)3   and  (x— y)2. 


Ans.  (a+6)6|3  and  (x— y)2|3. 
7.  Reduce  a3  and  65.  8.  Reduce  a;3  and  53. 

382.  When  it  is  required  to  reduce  a  quantity  to  a  given 
index ; 

Divide  the  index  of  the  quantity  by  the  given  index,  place 
the  quotient  over  the  quantity,  and  set  the  given  index  over 
the  whole. 

This  is  merely  resolving  the  original  index  into  two  factors, 
according  to  Art.  269. 

jl 
Ex.  1.    Reduce  a6  to  the  index  |. 

By  Art.  165,  i-i=iX^t  =  f 

This  is  the  index  to  be  placed  over  «,  which  then  becomes 

i  .         .  .  .       il2" 

a2  ;    and  the  given  index  set  over  this,  makes  it  a2\  ,  the 

answer. 

2.  Reduce  a2  and  x2  to  the  common  index  £. 

2-^-1=2X3=6,  the  first  index 
f-f-i=fX3  =  f,  the  second  index. 

1  9     1  .    .  . 

Therefore    (a6)'3  and  (x2)2  are  the  quantities  required. 
3.  Reduce  42  and  33,  to  the  common  index  }. 

Ans.  (4  s)*  and  (S2)K 

383.  Thirdly,  to  remove  a  part  of  a  root  from  under  the 
radical  sign ; 

If  the  quantity  can  be  resolved  into  two  factors,  one  of 
which  is  an  exact  power  of  the  same  name  with  the  root ; 


INVOLUTION      AND      EVOLUTION.  155 

-find  the  root  of  this  power,  and  prefix  it  to  the  other  factor, 
with  the  radical  sign  between  them. 

This  rule  is  founded  on  the  principle,  that  the  root  of  the 
product  of  two  factors  is  equal  to  the  product  of  their  roots. 
(Art.  270.) 

It  will  generally  be  best  to  resolve  the  radical  quantity  into 
such  factors  that  one  of  them  shall  be  the  greatest  power 
which  will  divide  the  quantity  without  a  remainder.  If  there 
is  no  exact  power  which  will  divide  the  quantity,  the  reduc- 
tion cannot  be  made. 

Ex.  1.  Remove  a  factor  from  ->/8. 
The  greatest  square  which  will  divide  8  is  4. 
We  may  then  resolve  8  into  the  factors  4  and  2.  For  4X2=8. 

The  root  of  this  product  is  equal  to  the  product  of  the 
roots  of  its  factors  ;   that  is,   «/8=  v/4X  \/2. 

But  \/4=2.  Instead  of  <s/4,  therefore,  we  may  substitute 
its  equal  2.     We  then  have  2X  >/2  or  2v/2. 

This  is  commonly  called  reducing  a  radical  quantity  to  its 
most  simple  terms.  But  the  learner  may  not  perhaps  at  once 
perceive,  that  2^/2  is  a  more  simple  expression  than   y/8. 

2.  Reduce   \^a2x.  Ans.   v/a2X\/x=aX^-fl\/^ 

3.  Reduce   v^is.  Ans.   ^9X2=^9X^2=3^2. 

4.  Reduce   V64&3c.  Ans.    v/6463X  yc=4b\/c. 

„  4  /a*b  a  4  /b 

5.  Reduce    V^-  Ans.  -\/~7*    (Art.  271.) 

6.  Reduce  Vanb.  Ans.  a\/b,  or  abn. 

7.  Reduce  (a3  —  a2b)2.  '       Ans.   a(a—b)2. 

8.  Reduce  (54a  6fc)3.  Ans.  3a2  (2b)*. 


9.  Reduce   ^98a2j\  10.  Reduce   Va*+a3b2. 

S84.  By  a  contrary  process,  the  co-efficient  of  a  radical 
quantity  may  be  introduced  under  the  radical  sign. 

1.  Thus,  ayb^Va^b. 

For  a—  yan  or  a*.      (Art.  264.)      And    yan  X  Vo=  Vanb. 


156  INVOLUTION      AND      EVOLUTION. 

Here  the  co-efficient  a  is  first  raised  to  a  power  of  the 
same  name  as  the  radical  part,  and  is  then  introduced  as  a 
factor  under  the  radical  sign. 


2.  a(x—b)*  =  (a3Xx—b)*  =  (a3x—a3b)*. 

3.  2ab(2ab2y=(l6a*b5)^. 


H    b*C    )*J    a%h*c    )* 

b\a2  +  b2/       \a2b2+b*/ 


Addition  and  Subtraction  op  Ra»ical  Quantities. 

S83>.  Radical  quantities  may  be  added  like  rational  quan- 
tities, by  writing  them  one  after  another  with  their  signs. 

Thus  the  sum  of  «/a  and  \/b}  is   </a+^/b. 

11  1         !  iiijl 

And  the  sum  of  a^—h*  and  x*—yn,  is  ar  —  h?+x*— yn. 

But  in  many  cases,  several  terms  may  be  reduced  to  one. 

The  sum  of  2^a  and  3</a  is  2</a+3%/a=5s/a. 

For  it  is  evident  that  twice  the  root  of  a,  and  three  times 
the  root  of  a,  are  five  times  the  root  of  a.     Hence, 

386.  When  the  quantities  to  be  added  have  the  same 
radical  part,  under  the  same  radical  sign  or  index ;  add  the 
rational  parts,  and  to  the  sum  annex  the  radical  parts. 

If  no  rational  quantity  is  prefixed  to  the  radical  sign,  1  is 
always  to  be  understood.     (Art.  255.) 


To 

Add 

2yay 
Vay 

SVay 

5^/a 
—2y/a 

3(a+A)T      5bhG 
4(x+h)*     7bh° 

aVb-h 
y^b—h 

Sum 

(a+y)  AVb-h 

287.  If  the  radical  parts  are  origin  all v  different,  they 
may  sometimes  be  made  alike,  by  th,»  i eductions  iji  the  pre- 
ceding articles. 

1.  Add  v/8  to  v/50.  Here  the  radical  parts  are  not  the 
same.  But  by  the  reduction  in  Art.  283,  ^8=2^2,  and 
^50^=5^2.     The  sum  then  is  7v/2. 


INVOLUTION      AND      EVOLUTION.  157 

2.  Add  v/16&  to  y/±b.  Ans.  4^6+2^6=6  >/b. 

3.  Add  y/a2x  to  x/&4;r.    Ans.  a  vAc-f  62  ^/x=(a+b2)X  y/x. 

4.  Add  (36a2y)i  to   (25y)*  Ans.  (6a+5)Xy*. 

5.  Add   x/18a  to  3v/2a. 

288.  But  if  the  radical  parts,  after  reduction,  are  differ- 
ent or  have  different  exponents,  they  cannot  be  united  in  the 
same  term ;  and  must  be  added  by  writing  them  one  after 
the  other. 

The  sum  of  S^b  and  2y/a,  is  3s/b+2y/a. 

It  is  manifest  that  three  times  the  root  of  b,  and  twice  the 
root  of  a,  are  neither  five  times  the  root  of  b,  nor  five  times 
the  root  of  a,  unless  b  and  a  are  equal. 

The  sum  of  %/a  and   \Va,  is    %/a+yVa. 

The  square  root  of  a,  and  the  cube  root  of  a,  are  neither 
twice  the  square  root,  nor  twice  the  cube  root  of  a. 

289.  Subtraction  of  radical  quantities  is  to  be  performed 
in  the  same  manner  as  addition,  except  that  the  signs  in  the 
subtrahend  are  to  be  changed  according  to  Art.  75. 

i 


From  y/ay       4^a-{-x  3h?       a(x+y)       —   a  n 

Subtract      3  y/ay       3va+x       — 5A3       b(x+y)       —2a  n 


Difference  -2  y/ay 

From  ^50,  subtract  y/Q.         Ans.  5^2—2^2=3^2. 
From  \fb*y,  subtract  l/by*.  Ans.  (b—y)X\Vby. 

From   y/x,  subtract  \Vx. 

Multiplication  of  Radical  Cjuantities. 

290.  Radical  quantities  may  be  multiplied,  like  other 
quantities,  by  writing  the  factors  one  after  another,  either 
with  or  without  the  sign  of  multiplication  between  them. 

Thus  the  product  of  y/a  into   y/b,  is   y/aX  y/b. 

i  i  i  i 

The  product  of  h*  into  y*  is  A3y^. 
14 


J  58  INVOLUTION      AND      EVOLUTION. 

But  it  is  often  expedient  to  bring  the  factors  under  the 
same  radical  sign.  This  may  be  done,  if  they  are  first  re- 
duced to  a  common  index. 

Thus  ny/xXVy—Vxy.  For  the  root  of  the  product  of 
several  factors  is  equal  to  the  product  of  their  roots.     Hence, 

£9  1 .  Quantities  under  the  same  radical  sign  or  index, 
may  be  multiplied  together  like  rational  quantities,  the  pro- 
duct being  placed  under  the  common  radical  sign  or  index. 

Multiply  %/x  into   yy,  that  is,  x^  into  y* 

The  quantities  reduced  to  the  same  index,  (Art.  281,)  are 

11  .  1       6/ 

(a:3)6,  and  (y2)6   and  their  product  is  (x3y2)6  =  vx3y2. 


i 


Multiply     s/a+m  ydx         a2  (a+y)n 


Into  \fa—m         </hy         x\         (b+h)'1 


Product       ^a2-m2  (a3x)2  (anxm)'mn 

Multiply   y/Sxb  into   V2xb.  Prod.    VlQx2b2=4xb. 

In  this  manner  the  product  of  radical  quantities  often  be- 
comes rational. 

Thus  the  product  of  -v/2  into   \/18=yS6=6. 

i  il 

And  the  product  of  (a2 y3) 4  into  (a2y)4:  —  (aiyi)±=ay. 

292.  Roots  of  the  same  letter  or  quantity  may  be  multi- 
plied, by  adding  their  fractional  exponents. 

The  exponents,  like  all  other  fractions,  must  be  reduced 
to  a  common  denominator,  before  they  can  be  united  in  one 
.term.     (Art.  152.) 

1  1  14-1  3    12  5_ 

Thus  a2  Xa3=a2^ 3  =a6^ 6  =a6 

The  values  of  the  roots  are  not  altered,  by  reducing  their 
indices  to  a  common  denominator.     (Art.  265.) 

Therefore  the  first  factor   a2=a*  ) 

And  the  second  a3=a*  ) 


INVOLUTION      AND      EVOLUTION.  159 

But  J=a^Xa^Xa^\     (Art.  261.) 

And  aT=a6X^. 

2         1         1         i         L         L 

The  product  therefore  is  a6  Xa6  Xa6  Xa6  Xa6  =a6 . 

And  in  all  instances  of  this  nature,  the  common  denomin- 
ator of  the  indices  denotes  a  certain  root ;  and  the  sum  of 
the  numerators,  shows  how  often  this  is  to  be  repeated  as  a 
factor  to  produce  the  required  product. 

Thus  cf  Xa™=a™nXamn=a~™: 

Multiply         3y*         <$Xc$         (a+b)%       (a-y)"     x~% 


A 


i 


Into  y*        a*  (a+b)*       {a-yY 


li. 


_7 


Product  3yT*  (*+*)*  x  lT 

The  product  of  y2  into  y  3,  is  y°     6=y6. 

J.  _x  -L__± 

The  product  of  an  into  a  n,  is  an    n==a°  =  l. 
And   xn^Xx^n=xn"n+^=x°  =  l. 

L         §.         1         1 

The  product  of  a2  into  a3=a3Xa*=a3. 

293.  From  the  last  example  it  will  be  seen,  that  powers 
and  roots  may  be  multiplied  by  a  common  rule.  This  is  one 
of  the  many  advantages  derived  from  the  notation  by  frac- 
tional indices.  Any  quantities  whatever  may  be  reduced  to 
the  form  of  radicals,  (Art.  280,)  and  may  then  be  subjected 
to  the  same  modes  of  operation. 

Thus    y3xy°=y3+lf=yJ°9~. 

X  1+-L  »+l 

And     xXxn=x     n=x  n  . 

The  product  will  become  rational,  whenever  the  numerator 
of  the  index  can  be  exactly  divided  by  the  denominator. 

1  2  1_2_ 

Thus    a3Xa?Xa*=d2  =a*. 

And      (a+byx(a+b)~^=(a+by=a+b. 

3  3  5 

And      a5Xa5~a5=a. 


160  INVOLUTION      AND      EVOLUTION. 

294.  When  radical  quantities  which  are  reduced  to  the 
same  index,  have  rational  co-efficients,  the  rational  parts  may 
be  multiplied  together,  and  their  product  prefixed  to  the  pro- 
duct of  the  radical  p arts. 

1.  Multiply  a^/b  into  Cy/d. 

The  product  of  the  rational  parts  is  ac. 
The  product  of  the  radical  parts  is   y/bd. 
And  the  whole  product  is  acy/bd. 
For  ay/b  is  aXy/b.     (Art.  255.)     And  Cy/d  is  cX^. 

By  Art.  96,  aX  y/b  into  cX  W,  is  aXVbXcXVd',  or 
by  changing  the  order  of  the  factors, 

aXcXy/bX  y/d=acX  y/bd=acy/bd. 

2.  Multiply   ax2  into  ba^. 

When  the  radical  parts  are  reduced  to  a  common  index, 
the  factors  become  a(x3)*    and  b(d2)6.  i. 

The  product  then  is  ab(x*d2)*. 

But  in  cases  of  this  nature  we  may  save  the  trouble  of 
reducing  to  a  common  index,  by  multiplying  as  in  Art.  290. 


1                1             11 
Thus  ax2   into  bd*,  is  ax%bd$. 

Multiply       a(b+x)^         ciy/y2       ay/x 

ax  £ 

xyz 

Into              y{b—xy         by/hy       by/x 

6y"l 

yVQ 

Product      ay(b2-x2)^  abs/x2=abx  Sxy 

295.  If  the  rational  quantities,  instead  of  being  co-effi- 
cients to  the  radical  quantities,  are  connected  with  them  by 
the  signs  +  and  — ,  each  term  in  the  multiplier  must  be 
multiplied  into  each  in  the  multiplicand,  as  in  Art.  95. 

Multiply  a+y/b 
Into  c+y/d 


ac+Cy/b 

a^/d+y/bd 

ac+Cy/b+ay/d+y/bd. 


INVOLUTION      AND      EVOLUTION.  161 

The  product  of  tt+v/y  into  l+ryy,  is  a+  yy+ aryy  +  ry. 

1.  Multiply   ya  into   yb.  Ans.   V^*b*. 

2.  Multiply  5v/5  into  3^8.  Ans.  30VlO. 

3.  Multiply  2^3  into  3^/4.  Ans.  6V432. 

4.  Multiply   v/c?  into    yah.  Ans.   Va2b2d3. 

5.  Multiply   V7^  into  \/^.         Ans.  >/***. 

1  J     V     3c  V     26  V       c 

6.  Multiply  a{a—xy   into   (c— c?)X(ax)^. 

Ans.  (ac— ad)X(a2z  —  ax2)*. 

Division  of  Radical  Quantities. 

296.   The  division  of  radical  quantities  may  be  expressed 

by  writing  the  divisor  under  the  dividend,  in  the  form  of  a 

fraction. 

ya 
Thus  the  quotient  of  ya  divided  by   yb,  is  — r- 

yo 

And  (a+hft  divided  by  (b+x)",  is  (g+A)^. 

In  these  instances,  the  radical  sign  or  index  is  separately 
applied  to  the  numerator  and  the  denominator.  But  if  the 
divisor  and  dividend  are  reduced  to  the  same  index  or  radical 
sign,  this  may  be  applied  to  the  whole  quotient. 

ya        n  /a 
Thus   ya-r-  V°—iri =  \r   r-     For  the  root  of  a  fraction 

is  equal  to  the  root  of  the  numerator  divided  by  the  root  of 
the  denominator,     (Art.  271.) 

Again,  yab+-  yb=  y~a.  For  the  product  of  this  quotient 
into  the  divisor  is  equal  to  the  dividend,  that  is, 

yaX  yb=  V ab.     Hence, 

29  ¥.  Quantities  under  the  same  radical  sign  or  index 
may  be  divided  like  rational  quantities,  the  quotient  being 
placed  under  the  common  radical  sign  or  index. 

Divide   (x*y2)J  by  y3. 

14* 


]62  INVOLUTION      AND      EVOLUTION. 

These  reduced  to  the  same  index  are  (x2y2)*  and  (ya)°  : 

13  1 

And  the  quotient  is   (x3)6=x*=x2. 


Divide         ^6a3x     ^  dhx2      (a3+ax)9       (a3h)m     (a*y2Y 
By  VSx         Vdx  a*  {ax)™       (ay)* 


Quotient     ^2a 3  (a2  +  x) 9  (aij) 4 

&98.  A  root  is  divided  by  another  root  of  the  same  letter 
or  quantity,  by  subtracting  the  index  of  the  divisor  from  that 
of  the  dividend. 

Thus   a'2~+d*=a?~°=a1>~~1>=a°=a3. 

j^  3         1         1         1  L 

For      a2=a6=a6XadXa6   and  this  divided  by  a6   is 

ill 

fl6Xfl6Xfl6        i       i       £       1 
r =a6Xa6=a6=a3. 


a6 
In  the  same  manner,  it  may  be  shown  that  am+an=a 


1  1  !_1 

TO        n 


!_f  2_  W-H»  2  1 

Divide      (3a)1?         (ax)3         a™  (b+y)^         (r2y2Y 

By  (3a)%  (axY         am  (b+y)"         (r2y2Y 


t 


Quotient  (3d)*  an  (r2y2)  T 

Powers  and  roots  may  be  brought  promiscuously  together, 
and  divided  according  to  the  same  rule.     See  Art.  293. 

Thus    a2  +  a*=a2~3  =  a*.      For   a*Xa3=a3=a2. 


»     m 


So    yn+ym=y 

299.  When  radical  quantities  which  are  reduced  to  the 
same  index  have  rational  co-efficients,  the  rational  parts  may 
be  divided  separately,  and  their  quotient  prefixed  to  the  quo- 
tient of  the  radical  parts. 

Thus  acy/bd+ay/b=c</d.  For  this  quotient  multiplied 
into  the  divisor  is  equal  to  the  dividend. 


INVOLUTION      AND      EVOLUTION  163 

Divide        2±x^ay     ISdh^/bx     by{a2x2)K     16v/32     by/xy 

x 
By  6   </a  2h<j  x      y(a  x  )n       8>/  4       y  y 

Quotient      4x^y  b(a2x  )*  &v/:c 

Divide    ab(x2b)^  by  a(x)2". 
These  reduced  to  the  same  index  are  ab(x2b)^  and  a(#*)* 

The  quotient  then  is  &(&)*=  (ft5)*      (Art.  284.) 

To  save  the  trouble  of  reducing  to  a  common  index,  the 
division  may  be  expressed  in  the  form  of  a  fraction. 

The  quotient  will  then  be    — ^ — ~ . 

a(x)? 

e  /b2 

1.  Divide    2ybc  by  Sy/ac.  Ans.  fv— • 

2.  Divide  10?/ 108  by  5%/ 4.  Ans.  22/27=6. 

3.  Divide  I0>/27  by  2^3.  Ans.  15. 

4.  Divide  8^108  by  2^6.  Ans.  12^2. 

5.  Divide  (a262a*3)*  by  cP.  Ans.  (afc)^. 

6.  Divide    (16a3  —  12a2 x)2  by  2a.         Ans.  (4a— 3x)¥. 

/ 

Involution  of  Radical  Quantities. 

300.  Radical  quantities,  like  powers,  are  involved  by 
multiplying  the  index  of  the  root  into  the  index  of  the  re- 
quired power. 

1.  The  square  of  a3=a^       =a3.     For  a3Xa2=a2. 

2.  The  cube  of  a^=a^XS=a^.     For  a?Xa?xd?=a%. 

3.  And  universally,  the  nth  power  of  am=am       =am.- 

1  -L  i. 

For  the  nth  power  of  am=amXam. . .  .n   times,    and  the 

n 

sum  of  the  indices  will  then  be   «. 


164  INVOLUTION      AND      EVOLUTION. 

JL    i  L    A. 

4.  The  5th  power  of  a2y3,  is  a2y3.  Or,  by  reducing 
the  roots  to  a  common  index, 

(a3y2)*X5  =  (a3y2)°. 

J.    J.  2L   M.  _?_ 

5.  The  cube  of  anxm,  is  anx**  or  (amxn)nm. 

mi  —     -  -     - 

6.  The  square  of  a3x*,  is  a3xT. 
The  cube  of  a*,  is  a^X3=«^=«. 

a 

J.  n. 

And  the  nth  power  of  an,  is  an=a.     That  is, 

301.  JL  roetf  is  raised  to  a  power  of  the  same  name,  by 
removing  the  index  or  radical  sign. 

Thus  the  cube  of  Vb-\-x,  is  b+x. 

And  the  rath  power  of  (a— i/)*,  is   (a—y). 

302.  When  the  radical  quantities  have  rational  co-effi- 
cients, these  must  also  be  involved. 

1.  The  square  of  atyx,  is  a2tyx2. 
For  atyxXatyx=a2  V/x2. 

-I  Vl 

2.  The  nth  power  of  amxm,  is  anmx™. 

3.  The  square  of  a^/x—y,  is  a2X(x—y)* 

4.  The  cube  of  3a2/ y,  is  27 a3 y. 

303.  But  if  the  radical  quantities  are  connected  with 
others  by  the  signs  +  and  — ,  they  must  be  involved  by  a 
multiplication  of  the  several  terms,  as  in  Art.  230. 

Ex.  1,  Required  the  squares  of  a+y/y  and  a—s/y. 

tt+  Vy  a— Vy 

aJr\/y  a— y/y 


a2+a^/y  a2—a^y 

aVy+y  -ay/y+y 


a2  +2ay/y+y  a2—2a^/y+y 

2.  Required  the  cube  of  a—  s/b. 

3.  Required  the  cube  of  2d+  y/x. 


INVOLUTION      AND      EVOLUTION.  1(15 

304.  It  is  unnecessary  to  give  a  separate  rule  for  the 
evolution  of  radical  quantities,  that  is,  for  finding  the  root  of 
a  quantity  which  is  already  a  root.  The  operation  is  the 
same  as  in  other  cases  of  evolution.  The  fractional  index  of 
the  radical  quantity  is  to  be  divided,  by  the  number  express- 
ing the  root  to  be  found  Or,  the  radical  sign  belonging  to 
the  required  root,  may  be  placed  over  the  given  quantity. 
(Art.  268.)  If  there  are  rational  co-efficients,  the  roots  of 
these  must  also  be  extracted. 

i  i— 2       I 

Thus,  the  square  root  of  a3,  is  a3  *     =a\ 

1  11 

The  cube  root  of  a(xy)2,  is  a3  (xy)*. 

The  rath  root  of  aVby,  is    v   aVby 

305*  It  may  be  proper  to  observe,  that  dividing  the  frac- 
tional index  of  a  root  is  the  same  in  effect,  as  multiplying  the 
number  which  is  placed  over  the  radical  sign.  For  this  num- 
ber corresponds  with  the  denominator  of  the  fractional  index  ; 
and  a  fraction  is  divided,  by  multiplying  its  denominator. 

i  i 

Thus    ya=<P  %/a=a«. 

i  -i- 

i/a=d*  *ya=a2n. 

On  the  other  hand,  multiplying  the  fractional  index  is 
equivalent  to  dividing  the  number  which  is  placed  over  the 
radical  sign. 

—    •  — X2       — 

Thus  the  square  of  %/a  or  a6,  is   %/a  or  a6       =a3. 

306.  In  algebraic  calculations,  we  have  sometimes  occa- 
sion to  seek  for  a  factor,  which  multiplied  into  a  given  radical 
quantity,  will  render  the  product  rational.  In  the  case  of  a 
simple  radical,  such  a  factor  is  easily  found.  For  if  the  nth 
root  of  any  quantity,  be  multiplied  by  the  same  root  raised 
to  a  power  whose  index  is  n—  1,  the  product  .will  be  the 
given  quantity. 

i       n~*       * 
Thus    yajXVaJ*"1    or  xnXx  *  =x"=x. 

i  n~1 

And    {x+y)nX(x+y)  *  =x+y. 

So    v/aX\/fl=fl.       And    V«X  \/aa  =  l/a3=a. 


166  INVOLUTION      AND      EVOLUTION. 

1  % 

And   VflXV«8  =  fl,  &c.     And  (a+6)^X(tf+&)3=a+&. 

1  3 

And  (x+y)*  X(x+y)*=x+y. 

307.  A  factor  which  will  produce  a  rational  product, 
when  multiplied  into  a  binomial  surd  containing  only  the 
square  root,  may  be  found  by  applying  the  principle,  that  the 
product  of  the  sum  and  difference  of  two  quantities,  is  equal 
to  the  difference  of  their  squares.  (Art.  111.)  The  binomial 
itself,  after  the  sign  which  connects  the  terms  is  changed 
from  +  to  — ,  or  from  —  to  +,  will  be  the  factor  required. 

Thus  (v^+\/&)X(\/a—  </b)=y/a2  —  ^b2=a— b,  which 
is  free  from  radicals. 

So      (l  +  x/2)X(l~vr2)  =  l-2=-l.  • 

And   (3-2v'2)x(3+2v/2)  =  l. 

When  the  compound  surd  consists  of  more  than  two  terms, 
it  may  be  reduced,  by  successive  multiplications,  first  to  a 
binomial  surd,  and  then  to  a  rational  quantity. 

Thus  (v'10-^2-x/3)X(v/10+v/24-v/3)==5~2v/6,  a 
binomial  surd. 

And  (5-2x/6)X(5+2v/6=l. 

Therefore  (\/10—  s/2  —  y/S)  multiplied  into  ( ^  10+ v/2 -f 
V/3)X(5+2V/6)  =  1. 

308.  It  is  sometimes  desirable  to  clear  from  radical  signs 
the  numerator  or  denominator  of  a  fraction.  This  may  be 
effected,  without  altering  the  value  of  the  fraction,  if  the 
numerator  and  denominator  be  both  multiplied  by  a  factor 
which  will  render  either  of  them  rational,  as  the  case  may 
require. 

\/a 
1.  If  both  parts  of  the  fraction  — —  be  multiplied  by   y/a, 

y/ x 

it  will  become  =  — — ,    in  which  the  numerator  is 

yxX  y/a      v  ax 
a  rational  quantity. 

Or  if  both  parts  of  the  given  fraction  be  multiplied  by  y/x, 
it  will  become   ,  in  which  the  denominator  is  rational. 


INVOLUTION      AND      EVOLUTION.  1C7 

$  $X(a+x)%      tfrxid+xft 

2.  The  fraction   -= f  = T~Uz  =  ~ — ~~~T 

(a+x)*       (a+x)^+^  a+x 


+  1 


3.  The  fraction   Vy-H^    (y+s)*     *=_J/+£_ 


a(y+x) 3  a(y+xYs 


n-\ 


4.  The  fraction 


a      ax  n  aVxn~1 

~~T        ~±  n- 1  ~~  ~ 

Xn        XnXX    n 


k    rru    r      *•  ^2  v/2X(3+x/2)         2+3^/2 

5.  The  fraction  ^-^  =  ^^—-^=  __. 

mi      ,        ,  3  3(x/5+v/2) 

#  6.  The  fraction  -^^-^— ^-^  ^5+^2. 

3 

7.  The  fraction  ^  =  <^  =  ?Vi25. 

8.  The  fraction 

8 8X(x/3-x/2-l)(-v/2) 

v/3+  ^2  +  1 ~~  (x/3+  %/2  +  lj(v'3-  v/2-l)(-  ^2) 
=  4-2v/6+2v/2. 

2 

9.  Reduce  — r  to  a  fraction  having  a  rational  denominator. 

10.  Reduce  — ; r  to  a  fraction  having;  a  rational  denom- 

a+>/b  ° 

inator. 

300.  In  a  similar  manner,  an  equation  may  be  cleared 
of  radicals  by  involution,  or  by  multiplying  both  sides  by 
such  factors  as  will  render  the  radical  quantities  rational. 

Ex.  1.  Clear  from  its  radical  sign  the  equation 

\/ac-\-d=x— ad. 
Squaring  both  sides, 

ac+d=x2  —2adx+a2d2. 

2.  Clear  from  radicals  the  equation 

Vb+a—  Va+x=y. 


168  INVOLUTION      AND      EVOLUTION. 

Transposing  and  squaring, 


b+a=y2+2yVa+x-\-a+x. 
Transposing  and  squaring  again, 

(b—y2  —x)2=4y2  X  (a+x). 

310*  The  arithmetical  operation  of  finding  the  proximate 
value  of  a  fractional  surd,  may  be  shortened,  by  rendering 
either  the  numerator  or  the  denominator  rational.  The  root 
of  a  fraction  is  equal  to  the  root  of  the  numerator  divided 
by  the  root  of  the  denominator.     (Art.  271.) 

Thus  \/|==  — •    But  this  may  be  reduced  to  Tj^n^l 

or  - r (A^.  308.) 

mi  a  .     y/a  a  y/aj, 

1  he  square  root  of   r  is  — r>    or   — =>    or   • 

-  b         Vb  Vab  b 

When  the  fraction  is  thrown  into  this  form,  the  process  of 
extracting  the  root  arithmetically,  will  be  confined  either  to 
the  numerator,  or  to  the  denominator. 

T,        .  '     ,  3       v/3       n/3Xv/7       v/21 

1  hus  the  square  root  of  -  =  — —  =    ,„w   — -  =  — r—  • 
"  7       V7       ^7X%/7         7 

Examples, 

1.  Find  the  4th  root  of  8 la2. 

2.  Find  the  6th  root  of  (a+b)' 3. 

3.  Fnd  the  nth  root  of  {x—yY. 


4.  Find  the  cube  root  of  —  125a:c8. 

4a* 

5.  Find  the  square  root  of 

6.  Find  the  5th  root  of 


dx2y2 
S2aBx10 


243 

7.  Find  the  square  root  of  x2  —  6bx+9b*. 

y2 

8.  Find  the  square  root  of  a*+ay+~- 

9.  Reduce  ax2  to  the  form  of  the  6th  root. 


INVOLUTION      AND      EVOLUTTON.  109 

10.  Reduce  —3y  to  the  form  of  the  cube  root. 

1 

11.  Reduce   a2   and  a'3  to  a  common  index. 

L  JL 

12.  Reduce  43  and  54  to  a  common  index. 

l  i 

13.  Reduce  a2  and  b*  to  the  common  index  £. 

i  i 

14.  Reduce  2^  and  4T  to  the  common  index  £. 

15.  Remove  a  factor  from   ^294. 


16.  Remove  a  factor  from   \^x3  —  a2x2. 

17.  Find  the  sum  and  difference  of    ^16a2x  and  \^4a2x. 

18.  Find  the  sum  and  difference  of    Vl92  and   V24. 

19.  Multiply  7^18  into  5^4. 

20.  Multiply  4+2v/2  into  2-^2. 

21.  Multiply  a(a-h</c)z  into  b(a—  x/c)2. 

22.  Multiply  2(a+6)"  into  3(a+Z>)™  . 

23.  Divide  6^54  by  3^2. 

24.  Divide  4V72  by  2V1& 

25.  Divide   y/1  by   VI. 

26.  Divide  8>/512  by  4V2. 

27.  Find  the  cube  of  17^21. 

28.  Find  the  square  of  5-rV2. 

29.  Find  the  4th  power  of  \  ^/6. 

30.  Find  the  cube  of  x/x—  y/b. 

31.  Find  a  factor  which  will  make   \/y  rational. 

32.  Find  a  factor  which  will  make   \f 5—  y/x  rational. 

33.  Reduce  to  a  fraction  having;  a  rational  numerator. 

y/X  b 

34.  Reduce  to  a  fraction  having  a  rational  d^- 

nominator. 

15 


170  aUADRATIC      EQUATIONS. 


SECTION    X. 


REDUCTION    OF    EQUATIONS    BY    INVOLUTION    AND 
EVOLUTION. 

Art.  311.  In  an  equation,  the  letter  which  expresses  the 
unknown  quantity  is  sometimes  found  under  a  radical  sign. 
We  may  have  ^/x  =  a. 

To  clear  this  of  the  radical  sign,  let  each  member  of  the 
equation  be  squared,  that  is,  multiplied  into  itself.  We  shall 
then  have 

^xX>/x=aa  Or,  (Art.  301,)     x=aa. 

The  equality  of  the  sides  is  not  affected  by  this  operation, 
because  each  is  only  multiplied  into  itself,  that  is,  equal  quan- 
tities are  multiplied  into  equal  quantities. 

The  same  principle  is  applicable  to  any  root  whatever. 
If  yX—a\  then  x=an.  For  by  Art.  1301,  a  root  is  raised  to 
a  power  of  the  same  name,  by  removing  the  index  or  radical 
sign.     Hence, 

31^.  When  the  unknown  quantity  is  under  a  radical 
sign,  the  equation  is  reduced  by  involving  both  sides,  to  a 
power  of  the  same  name,  as  the  root  expressed  by  the  radical 
sign. 

It  will  generally  be  expedient  to  make  the  necessary  trans- 
positions, before  involving  the  quantities ;  so  that  all  those 
which  are  not  under  the  radical  sign  may  stand  on  one  side 
of  the  equation. 

Ex.  1.  Reduce  the  equation  ^#+4=9 

Transposing  +4  %/x=9— 4  =  5 

Involving  both  sides,  x=52  =25. 

2.  Reduce  the  equation  a-\-\/x—b= d 

By  transposition,  \/x=d-\-b—a 

By  involution,  x=(d+b  —  a)\ 


aUADRATl 

C      EQUATIONS.                              171 

3.  Reduce  the  equation 
Involving  both  sides, 
And 

V^+T=4 

x+l=43=64 
#=63 

4.  Reduce  the  equation 
Clearing  of  fractions, 

4+3^^-4=6+4 

8+6^-4=13 

And 

Involving  both  sides, 

And 

^-4=1- 
*-4=tf 

2  +  ,/ 

5.  Reduce  the  equation 

>/ a2  -4-   /t  " *w 

y/aa  +  y/x    . 

Multiplying  by   V a2  ■ 

And 

Involving  both  sides, 

f^ 

</x=3+d—a2 
x=(S+d-a2)2. 

In  the  first  step  in  this  example, 

multiplying  the  first  mem- 

ber  into  ^ a2  +  y/x,  that  is,  into  itself,  is  the  same  as  squaring 
it,  which  is  done  by  taking  away  its  radical  sign.  The  other 
member  being  a  fraction,  is  multiplied  into  a  quantity  equal 
to  its  denominator,  by*cancelling  the  denominator.  (Art.  162.) 
There  remains  a  radical  sign  over  x,  which  must  be  removed 
by  involving  both  sides  of  the  equation. 

6.  Reduce  3+2^/x— j=6.  Ans.  3=Hf 

7.  Reduce  4\/f  =8.  Ans.  z=20. 

v     5 

8.  Reduce  (2#+3)3+4=7.  Ans.  z=12. 


9.  Reduce   ^12+x—  2+s/x.            •  Ans.  x=4. 

, ,  25a 

10.  Reduce   Vx— a=  y/x— \<Ja.  Ans.  x=—* 

9 

11.  Reduce   v/5X  ^+2=2+  \/$x.  Ans.  ^=^' 

„    .         x—ax      y/x  .                  1 

12.  Reduce   = Ans.  x=- — -• 

v/x          x  1— « 

„    i          ^x+28       v/x+38.  A               4 

13.  Reduce  — —  =  — -*.  Ans.  x=4. 


172  QUADRATIC      EQUATIONS. 

. 2a 

14.  Reduce   y/x+va-\-x=-—=-  Ans.  x=\a. 

Va+x 

2a2 


15.  Reduce  x+  V 'a2  +x2  =  — Ans.  x=«v/i. 

Va2+x2 

b2-4a2 


16.  Reduce  #+£=  ^a2 -\-Xy/02  ^.X2.     Ans.  #= 


4a 


2 


17.  Reduce   ^2+x-\-  */x= — .  Ans.  x=-- 

V2+x  3 


18.  Reduce   ^x  — 32=16—  y/x.  Ans.  ar=81 


jg   19.  Reduce   ^4^+17=2^^  +  1.  Ans.  x=16. 

„    ,  ^6x-2      4^to-9 

20.  Reduce   ■— ■== = — ^=- Ans.  ar=6. 

^Gx+2      4^6x+6 

Reduction  of  Equations  by  Evolution. 

313.  In  many  equations,  the  letter  which  expresses  the 
unknown  quantity  is  involved  to  some  power.  Thus  in  the 
equation  X*  =  16 

we  have  the  value  of  the  square  of  x,  but  not  of  x  itself. 
If  the  square  root  of  both  sides  be  extracted,  we  shall  have 

x=4. 

The  equality  of  the  members  is  not  affected  by  this  reduc- 
tion. For  if  two  quantities  or  sets  of  quantities  are  equal, 
their  roots  are  also  equal. 

If  (x  +  a)n=b+h,   then  x+a=Vb+h.     Hence, 

314.  When  the  expression  containing  the  unknown  quan- 
tity is  a  power,  the  equation  is  reduced  by  extracting  the  root 
of  both  sides,  a  root  of  the  same  name  as  the  power. 

Ex.  1.  Reduce  the  equation  6+x2—  8=7 

By  transposition,  # 2  =  7  +  8  —  6=9 

By  evolution,  #  =  =h\/9=zh3. 

The  signs  +  and  —  are  both  placed  before  v/9,  because  an 
even  root  of  an  affirmative  quantity  is  ambiguous.  (Art.  272.) 


aUADRATIC      EQUATIONS.  173 

2.  Reduce  the  equation  5x2  —  30=xa-f  34 
Transposing,  &c.  x2  =  \6 

By  evolution,  x=db4. 

X2  x2 

3.  Reduce  the  equation    a  +  -j-=h j 

rt%      .         r  r       -  o  n     hdh  —  abd 

Clearing  of  fractions,  &c.    x2  = — 7  ,    , — 

b+d 

~  .     .  /bdh  —  abd\i 

By  evolution,  x=±:\ — ,       —  1  • 

4.  Reduce  the  equation  a+dxn  =10— xn 

Transposing,  &c,  xn=   ,       • 

,     •  /10— a\i 

By  evolution,  ^=  wXT/  * 

31«>«  From  the  preceding  articles,  it  will  be  easy  to  see 
in  what  manner  an  equation  is  to  be  reduced,  when  the  ex- 
pression containing  the  unknown  quantity  is  a  power,  and  at 
the  same  time  under  a  radical  sign ;  that  is,  when  it  is  a  root 
of  a  power.  Both  involution  and  evolution  will  be  necessary 
in  this  case. 

Ex.  1.  Reduce  the  equation  %/x2=4. 

By  involution,  x2=43=64: 

By  evolution,  x=zhy/64=zhS. 

2.  Reduce  the  equation  >/xm  —  a=h  —  d 

By  involution,  xm  —  a=h2—2hd-\-d2 

And  xm=h2  -2hd+d2  +a 


By  evolution,  x=^h2  -2hd+d2  +a. 

3.  Reduce  the  equation    (x+aYI= j 

(x-a)* 

Multiplying  by  (x—a)^    (Art.  291,)    (x2-a2)^=a+b 
By  involution,  x2  —  a2=a2 -\-2ab-\-b* 

Transposing  and  uniting  terms,  x2=2a2  +2ab+b2 

By  evolution,  x=  (2a2  +2ab+b2)*. 

15* 


174  ClUADRATIC      EaUATIONS. 


Problems. 

Prob.  1.  A  gentleman  being  asked  his  age,  replied,  "If 
you  add  to  it  ten  years,  and  extract  the  square  root  of  the 
sum,  and  from  this  root  subtract  2,  the  remainder  will  be  6." 
What  was  his  age  ? 


By  the  conditions  of  the  problem  */x-\- 10—2  =  6 

By  transposition,  ^£+10=6+2=8 
By  involution,  a:+10=82=64 

And  x=  64  -10=54 

Proof  (Art.  197,)  V 54  +  10-2=6. 

Prob.  2.  If  to  a  certain  number  22577  be  added,  and  the 
square  root  of  the  sum  be  extracted,  and  from  this  163  be  sub- 
tracted, the  remainder  will  be  237.     What  is  the  number  ? 

Let  x=  the  number  sought  6=163 

«=22577  c=237. 

By  the  conditions  proposed,  ^x+a  —  b=c 

By  transposition,  ^x-\-a=c-{-b 

By  involution,  x-\-a~(c-\-b)2 

And  x=(c+b)2—  a 

Restoring  the  numbers,  (Art.  47,)  x  =  (237  + 1 63) 2  —  22577 
That  is,  x— 160000— 22577=  137423. 

Proof     ^137423+22577- 163=237. 

316.  When  an  equation  is  reduced  by  extracting  an  even 
root  of  a  quantity,  the  solution  does  not  determine  whether 
the  answer  is  positive  or  negative.  (Art.  314.)  But  what 
is  thus  left  ambiguous  by  the  algebraic  process,  is  frequently 
settled  by  the  statement  of  the  problem. 

Prob.  3.  A  merchant  gains  in  trade  a  sum,  to  which  320 
dollars  bears  the  same  proportion  as  five  times  the  sum  does 
to  2500.     What  is  the  amount  gained  ? 

Let  x=  the  sum  required. 
a=320. 
6=2500. 


aUADRATIC      EQUATIONS.  175 

By  the  supposition,  a  \  x\  \5x  :  b 

Multiplying  the  extremes  and  means,         5x2=ab 

(ab\l 
And  *-y* 

^                               ,                              /320X2500U 
Restoring  the  numbers,  x~\ 1  =400. 

Here  the  answer  is  not  marked  as  ambiguous,  because  by 
the  statement  of  the  problem  it  is  gain,  and  not  loss.  It 
must  therefore  be  positive.  This  might  be  determined,- in  the 
present  instance,  even  from  the  algebraic  process.  When- 
ever the  root  of  x2  is  ambiguous,  it  is  because  we  are  igno- 
rant whether  the  power  has  been  produced  by  the  multiplica- 
tion of  +x,  or  of  —x,  into  itself.  (Art.  273.)  But  here  we 
have  the  multiplication  actually  performed.  By  turning  back 
to  the  two  first  steps  of  the  equation,  we  find  that  5x2  was 
produced  by  multiplying  5x  into  x,  that  is   +5x  into  +x. 

P?*ob.  4.  The  distance  to  a  certain  place  is  such,  that  if 
96  be  subtracted  from  the  square  of  the  number  of  miles,  the 
remainder  will  be  48.     What  is  the  distance  ? 

Let  x—   the  distance  required. 
By  the  supposition,  x2  —  96=48 

Therefore,  x=  s/ 144= 12. 

Prob.  5.  If  three  times  the  square  of  a  certain  number  be 
divided  by  four,  and  if  the  quotient  be  diminished  by  12,  the 
remainder  will  be  180.     What  is  the  number? 

Sx2 

By  the  supposition,  — 12=180. 

Therefore,  x=  </256=  16. 

Prob.  6.  What  number  is  that,  the  fourth  part  of  whose 
square  being  subtracted  from  8,  leaves  a  remainder  equal  to 
four  ?  Ans.  4. 

Prob.  7.  What  two  numbers  are  those,  whose  sum  is  to 
the  greater  as  10  to  7;  and  whose  sum  multiplied  into  the 
less  produces  270  ? 

Let  10#=  their  sum. 

Then  lx—  the  greater,  and  Sx=  the  less. 

Therefore  x=S,  and  the  numbers  required  are  21  and  9. 


176  QUADRATIC      EQUATIONS. 

P?-ob.  8.  What  two  numbers  are  those,  whose  difference 
is  to  the  greater  as  2:9,  and  the  difference  of  whose 
squares  is  128?  Ans.  18  and  14. 

Prob.  9.  It  is  required  to  divide  the  number  18  into  two 
such  parts,  that  the  squares  of  those  parts  may  be  to  each 
other  as  25  to  16. 

Let  x—  the  greater  part.  Then  18— x=  the  less. 

By  the  condition  proposed,  x2  \  (18— x)2\  !25  :  16 

Therefore,  16#3=25X(18-#)» 
By  evolution,  4#=5X(18— x) 

And  #=10. 

Prob.  10.  It  is  required  to  divide  the  number  14  into  two 
such  parts,  that  the  quotient  of  the  greater  divided  by  the 
less,  may  be  to  the  quotient  of  the  less  divided  by  the  greater, 
as  16  !  9.  Ans.   The  parts  are  8  and  6. 

Prob.  11.  What  two  numbers  are  as  5  to  4,  the  sum  of 
whose  cubes  is  5103? 

Let  5x  and  4x=  the  two  numbers. 

Then  #=3,  and  the  numbers  are  15  and  12. 

Prob.  12.  Two  travellers  A  and  B  set  out  to  meet  each 
other,  A  leaving  the  town  C,  at  the  same  time  that  B  left  D. 
They  travelled  the  direct  road  between  C  and  D ;  and  on 
meeting,  it  appeared  that  A  had  travelled  18  miles  more  than 
B,  and  that  A  could  have  gone  J5's  distance  in  15|  days,  but 
B  would  have  been  28  days  in  going  A's  distance.  Required 
the  distance  between  C  and  D. 

Let  x—  the  number  of  miles  A  travelled. 
Then  x— 18=  the  number  B  travelled. 

3-  =A  s  daily  progress. 

x 

—  =J5's  daily  progress. 

™,        n  #—18      x 

Therefore,    x  :  #— 18!  ;— -=-  :  ~. 
15£       28 

*  This  reduced  gives  #=72,  A's  distance. 

The  whole  distance,  therefore,  from  C  to  D=  126  miles. 


QUADRATIC      EQUATIONS.  177 

Prob.  13.  Find  two  numbers  which  are  to  each  other  as 
8  to  5,  and  whose  product  is  360.  Ans.   24  and  15. 

Prob.  14.  A  gentleman  bought  two  pieces  of  silk,  which 
together  measured  36  yards.  Each  of  them  cost  as  many 
shillings  by  the  yard  as  there  were  yards  in  the  piece,  and 
their  whole  prices  were  as  4  to  1.  What  were  the  lengths 
of  the  pieces  ?  Ans.   24  and  12  yards. 

Prob.  15.  Find  two  numbers  which  are  to  each  other  as 
3  to  2 ;  and  the  difference  of  whose  fourth  powers  is  to  the 
sum  of  their  cubes,  as  26  to  7. 

Ans.    The  numbers  are  6  and  4. 

Prob.  16.  Several  gentlemen  made  an  excursion,  each 
taking  the  same  sum  of  money.  Each  had  as  many  servants 
attending  him  as  there  were  gentlemen ;  the  number  of  dol- 
lars which  each  had  was  double  the  number  of  all  the  serv- 
ants, and  the  whole  sum  of  money  taken  out  was  3456  dollars. 
How  many  gentlemen  were  there  ?  Ans.  12. 

Prob.  17.  A  detachment  of  soldiers  from  a  regiment  being 
ordered  to  march  on  a  particular  service,  each  company  fur- 
nished four  times  as  many  men  as  there  were  companies  in 
the  whole  regiment ;  but  these  being  found  insufficient,  each 
company  furnished  three  men  more  ;  when  their  number  was 
found  to  be  increased  in  the  ratio  of  17  to  16.  How  many 
companies  were  there  in  the  regiment  ?  Ans.  12. 

Affected  Quadratic  Equations. 

317.  Equations  are  divided  into  classes,  which  are  dis- 
tinguished from  each  other  by  the  power  of  the  letter  that 
expresses  the  unknown  quantity.  Those  which  contain  only 
theirs*  power  of  the  unknown  quantity  are  called  equations 
of  one  dimension,  or  equations  of  the  first  degree.  Those 
in  which  the  highest  power  of  the  unknown  quantity  is  a 
square,  are  called  quadratic,  or  equations  of  the  second, 
degree;  those  in  which  the  highest  power  is  a  cube,  equations 
of  the  third  degree,  &c. 

Thus  x—a+b,  is  an  equation  of  the  first  degree. 

x2=c,  and  x2+ax=d,  are  quadratic  equations,  or 
equations  of  the  second  degree. 

x3—h,  and  x3+ax2 +bx=d,  are  cubic  equations, 
or  equations  of  the  third  degree. 


178  QUADRATIC      EQUATIONS. 

318.  Equations  are  also  divided  into  pure  and  affected 
equations.  A  pure  equation  contains  only  one  power  of  the 
unknown  quantity.  This  may  be  the  first,  second,  third,  or 
any  other  power.  An  affected  equation  contains  different 
powers  of  the  unknown  quantity.     Thus, 

C  x2=d—b,  is  a  pure  quadratic  equation. 
I  x2  +bz=d,  an  affected  quadratic  equation. 
c  xz  —  b  —  c,  a  pure  cubic  equation. 
C  x3+ax2  +bx=h,  an  affected  cubic  equation. 

A  pure  equation  is  also  called  a  simple  equation.  But  this 
term  has  been  applied  in  too  vague  a  manner.  By  some 
writers,  it  is  extended  to  pure  equations  of  every  degree ;  by 
others,  it  is  confined  to  those  of  the  first  degree. 

In  a  pure  equation,  all  the  terms  which  contain  the  un- 
known quantity  may  be  united  in  one,  (Art.  189,)  and  the 
equation,  however  complicated  in  other  respects,  may  be 
reduced  by  the  rules  which  have  already  been  given.  But 
in  an  affected  equation,  as  the  unknown  quantity  is  raised  to 
different  powers,  the  terms  containing  these  powers  cannot 
be  united.  (Art.  244.)  There  are  particular  rules  for  the 
reduction  of  quadratic,  cubic,  and  biquadratic  equations. 
Of  these,  only  the  first  will  be  considered  at  present. 

319.  An  affected  quadratic  equation  is  one  which  con- 
tains  the  unknown  quantity  in  one  term,  and  the  square  of 
that  quantity  in  another  term. 

The  unknown  quantity  may  be  originally  in  several  terms 
of  the  equation.  But  all  these  may  be  reduced  to  two,  one 
containing  the  unknown  quantity,  and  the  other  its  square. 

3^4>.  It  has  already  been  shown  that  a  pure  quadratic  is  . 
solved  by  extracting  the  root  of  both  sides  of  the  equation. 
An  affected  quadratic  may  be  solved  in  the  same  way,  if  the 
member  which  contains  the  unknown  quantity  is  an  exact 
square.     Thus  the  equation 

x2+2ax+a2=b+h, 

may  be  reduced  by  evolution.     For  the  first  member  is  the 
square  of  a  binomial  quantity.  (Art.  276.)  And  its  root  is  x  +  a, 

Therefore,     x+a  —  y/b+h,  and  by  transposing  a, 

x—  Vb+h  —  a. 


QUADRATIC      EQUATIONS.  179 

'121.  But  it  is  not  often  the  case,  that  a  member  of  ail 
affected  quadratic  equation  is  an  exact  square,  till  an  addi- 
tional term  is  supplied,  for  the  purpose  of  making  the  required 
reduction.     In  the  equation 

x2  +2ax=b, 

the  side  containing  the  unknown  quantity  is  not  a  complete 
square.  The  two  terms  of  which  it  is  composed  are  indeed 
such  as  might  belong  to  the  square  of  a  binomial  quantity. 
But  one  term  is  wanting.  We  have  then  to  inquire,  in  what 
way  this  may  be  supplied.  From  having  two  terms  of  the 
square  of  a  binomial  given,  how  shall  we  find  the  third? 

Of  the  three  terms,  two  are  complete  powers,  and  the  other 
is  twice  the  product  of  the  roots  of  these  powers  ;  (Art.  109,) 
or  which  is  the  same  thing,  the  product  of  one  of  the  roots 
into  twice  the  other.     In  the  expression 

x2-\-2ax, 

the  term  2ax  consists  of  the  factors  2a  and  x.  The  latter 
is  the  unknown  quantity.  The  other  factor  2a  may  be  con- 
sidered the  co-efficient  of  the  unknown  quantity;  a  co-efficient 
being  another  name  for  a  factor.  As  x  is  the  root  of  the 
first  term  x2  ;  the  other  factor  2a  is  twice  the  root  of  the 
third  term,  which  is  wanted  to  complete  the  square.  There- 
fore half  2a  is  the  root  of  the  deficient  term,  and  a2  is  the 
term  itself.     The  square  completed  is 

x2  -\-2ax-\-a2 , 
m 
where  it  will  be  seen  that  the  last  term  a2   is  the  square  of 
half  2a,  and  2a  is  the  co-efficient  of  x,  the  root  of  the  first 
term. 

In  the  same  manner,  it  may  be  proved,  that  the  last  term 
of  the  square  of  any  binomial  quantity;  is  equal  to  the  square 
of  half  the  co-efficient  of  the  root  of  the  first  term.  From 
this  principle  is  derived  the  following  rule : 

*$^£l.  To  complete  the  square  in  an  affected  quadratic 
equation  :  take  the  square  of  half  the  co-efficient  of  the  first 
power  of  the  unknown  quantity,  and  add  it  to  both  sides  of 
the  equation. 

Before  completing  the  square,  the  known  and  unknown 
quantities  must  be  brought  on  opposite  sides  of  the  equation 
by  transposition;    and  the  highest  power  of  the  unknown 


180  QUADRATIC      EQUATIONS. 

quantity  must  have  the  affirmative  sign,  and  be  cleared  of 
fractions,  co-efficients,  &c.     See  Arts.  325,  6,  7,  8. 

After  the  square  is  completed,  the  equation  is  reduced,  by- 
extracting  the  square  root  of  both  sides,  and  transposing  the 
known  part  of  the  binomial  root.     (Art.  320.) 

The  quantity  which  is  added  to  one  side  of  the  equation, 
to  complete  the  square,  must  be  added  to  the  other  side  also, 
to  preserve  the  equality  of  the  two  members.     (Ax.  1.) 

323.  It  will  be  important  for  the  learner  to  distinguish 
between  what  is  peculiar  in  the  reduction  of  quadratic  equa- 
tions, and  what  is  common  to  this  and  the  other  kinds  which 
have  already  been  considered.  The  peculiar  part,  in  the 
resolution  of  affected  quadratics,  is  the  completing  of  the 
square.  The  other  steps  are  similar  to  those  by  which  pure 
equations  are  reduced. 

For  the  purpose  of  rendering  the  completing  of  the  square 
familiar,  there  will  be  an  advantage  in  beginning  with  exam- 
ples in  which  the  equation  is  already  prepared  for  this  step. 

Ex.  1.  Reduce  the  equation  x2  -\-6ax—b 

Completing  the  square,       x2  -\-6ax-\-9a2  —9a2  +b 


Extracting  both  sides,  (Art.  320,)  x+3a  =  ±:  V9a2  +b 

And  x=  —  Sado^9aa+b. 

Here  the  co-efficient  of  x,  in  the  first  step,  is  6a ; 

The  square  of  half  this  is  9a2,  which  being  added  to  both 
sides  completes  the  square.  The  equation  is  then  reduced 
by  extracting  the  root  of  each  member,  in  the  same  manner 
as  in  Art.  314,  excepting  that  the  square  here  being  that  of 
a  binomial,  its  root  is  found  by  the  rule  in  Art.  277. 

2.  Reduce  the  equation  x2  —Sbx=h 

Completing  the  square,  x2  —  Sbx  +  I6b2  =  16b2  +h 

Extracting  both  sides,  x  —4b=dz^l6b2+h 

And  x=4b±\/16b2+h. 

In  this  example,  half  the  co-efficient  of  x  is  4b,  the 
square  of  which  16b2  is  to  be  added  to  both  sides  of  the 
equation. 


aUADRATIC      EaUATIONS. 

x2+ax=b+h 


181 


3.  Reduce  the  equation 
Completing  the  square, 

By  evolution, 

And 

4.  Reduce  the  equation 
Completing  the  square, 

And 


a2      a2      1     t 
x2+ax-\~-r  =  —+b+h 
4        4 

x2—x—h—d 
x2-x+l=%+h-d 


Here  the  co-efficient  of  x  is  1,  the  square  of  half  which 


» i. 


5.  Reduce  the  equation 
Completing  the  square, 
And 


x2+3x=d+6 

x2+3x+%  =  %+d+6 


6.  Reduce  the  equation         x2—abx=ab—cd 

#2^2      a2b2 
Completing  the  square,  x2  —abx-\ — - — =  — — Yab—cd 


ab     (a2b2        7         \i 
x=—  ±[—  +ab-cd)  ' 


is 


And 

7.  Reduce  the  equation 

Completing  the  square, 

And 
By  Art.  161,  ~rz==~}^x'     The  co-efficient  of  x,  therefore, 

CL  CL  CI2 

t-     Half  of  this  is  -r>   the  square  of  which  is  -tj- 
16 


ax     i 
x2  +  -r=h 
b 

ax       a2        a2 
a      /a2       ,U 


1 82  QUADRATIC      EQUATIONS. 


8.  Reduce  the  equation 

**~l=*fh 

Completing  the  square, 

x        1          1 

And 

x=k4^+"hf- 

Here  the  fraction  t=tX^.  Therefore  the  co-efficient  of  x  is  - 
b     b  b 

324.  In  these  and  similar  instances,  the  root  of  the  third 
term  of  the  completed  square  is  easily  found,  because  this 
root  is  the  same  half  co-efficient  from  which  the  term  has 
just  been  derived.     (Art.  322.)     Thus  in  the  last  example, 

half  the  co-efficient  of  x  is  —r>   and  this  is  the  root  of  the 

Zo 

third  term    -7-. 
4b2 

325.  When  the  first  power  of  the  unknown  quantity  is 
in  several  terms,  these  should  be  united  in  one,  if  they  can 
be  by  the  rules  for  reduction  in  addition.  But  if  there  are 
literal  co-efficients,  these  may  be  considered  as  constituting, 
together,  a  compound  co-efficient  or  factor,  into  which  the 
unknown  quantity  is  multiplied. 

Thus  ax-\-bx+dx=(a+b+d)Xx.  (Art.  119.)  The  square 
of  half  this  compound  co-efficient  is  to  be  added  to  both 
sides  of  the  equation. 

1.  Reduce  the  equation  x2  +Sx+2x+x=d 
Uniting  terms,                       x2-\-6x  —  d 
Completing  the  square,         x2+Gx+9=9  +  d 
And                                         x=  —  3±V9-M. 

2.  Reduce  the  equation  x2  +ax+bx=h 
By  Art.  119,  x2 +  (a+b)Xx=h 

(aA-h\2      la-\-b\2 
~2/       \~2~7   *h 


By  evolution,  x+^=±\/(^±y+h 

And  *=— ^V  (-3-)  +*• 


aUADRATIC      EaUATIONS.  183 

3.  Reduce  the  equation     x2-\-ax  — x=b 
By  Art.  119,     x2+(a-l)Xx=b 

Therefore,  x2  +  (a- 1)  X^+^j2  =  r~^j  +6 

326.  After  becoming  familiar  with  the  method  of  com- 
pleting the  square,  in  affected  quadratic  equations,  it  will  be 
proper  to  attend  to  the  steps  which  are  preparatory  to  this. 
Here,  however,  little  more  is  necessary,  than  an  application 
of  rules  already  given.  The  known  and  unknown  quantities 
must  be  brought  on  opposite  sides  of  the  equation  by  trans- 
position. And  it  will  generally  be  expedient  to  make  the 
square  of  the  unknown  quantity  the  first  or  leading  term,  as 
in  the  preceding  examples.  This  indeed  is  not  essential. 
But  it  will  show,  to  the  best  advantage,  the  arrangement  of 
the  terms  in  the  completed  square. 

1.  Reduce  the  equation  a+5z— 3b=3x—x2 

Transposing  and  uniting  terms,  x2  -\-2x=3b—a 
Completing  the  square,  x2  +2x+l  =  l-\-3b—a 

And  x=-l±:^l+3b-a. 

x       36 


2.  Reduce  the  equation 


2      x+2 

Clearing  of  fractions,  &c.  x2  +  10.r=56 
Completing  the  square,     x2  +  10^+25=25+56=81 
And  #=-5dbv/81  =  -5db9. 

327.  If  the  highest  power  of  the  unknown  quantity  has 
any  co-efficient,  or  divisor,  it  fnust,  before  the  square  is  com- 
pleted, by  the  rule  in  Art.  322,  be  freed  from  these,  by  multi- 
plication or  division,  as  in  Arts.  184  and  188. 

1.  Reduce  the  equation  x2 -\-24a  —  6/*=l 2x— bx2 

Transposing  and  uniting  terms,  6x2  —  I2x=6h--24a 
Dividing  by  6,  x2  —  2x=h— 4.a 

Completing  the  square,  x2  —  2x+ 1  =  1  +h — 4a 

Extracting  and  transposing,       x=l±^l+h— 4a. 


184  QUADRATIC      EQUATIONS. 

bx2 


2.  Reduce  the  equation         h+2x=d- 


a 


Clearing  of  fractions,        bx2 -\-2ax= ad— ah 

2ax      ad—ah 
Dividing  by  b,  x2  +  -j-  = — 7 — 

2ax      a2      a2      ad— ah 
Therefore,  x2  +  -j-  +  y2  =  ^  +  — — 

a      (a2      ad—ah\k 
And  x=-T±  \-n:+- 


T7/ 


328.  If  the  square  of  the  unknown  quantity  is  in  several 
terms,  the  equation  must  be  divided  by  all  the  co-efficients 
of  this  square,  as  in  Art.  189. 

1.  Reduce  the  equation  bx2  +dx2  —4x=b—h 

4.x        b — h 
Dividing  by  b+d,  (Art.  124,)  **_  —  =  — 


Therefore,  x—  =— -=  ±  V/  ( 1 

b+d       v    \b+d) 


2     b-h 

\b+dJ  +b+d' 

2.  Reduce  the  equation  ax2+x=h-\-3x— x2 

Transp.  and  uniting  terms,  ax2  +x2  —2x=h 

2x  h 

Dividing  by  a+1,  *  x2 — -  =  — — - 

0    J  a+1      a+1 

Comp.  the  square,  x2 -—-+[——)  =(-77)  +  - 
a+l      \a+l/       \a+l/       a 


+  1 


Extracting  and  transp.  x= db  S/  I )  -j — . 

0+1  \a  +  l/       a  +  l 

There  is  another  method  of  completing  the  square,  which, 
in  many  cases,  particularly  those  in  which  the  highest  power 
of  the  unknown  quantity  has  a  co-efficient,  is  more  simple  in 
its  application,  than  that  given  in  Art.  322. 

Let  ax2+bx=d. 

If  the  equation  be  multiplied  by  4a,  and  if  b2  be  added 
to  both  sides,  it  will  become 

4a2x2+4abx+b2=4ad+b2  ; 
the  first  member  of  which  is  a  complete  power  of  2ax+6. 


aUADRATIC      EaUATIONS.  185 

Hence, 

?$20.  In  a  quadratic  equation,  the  square  may  be  com- 
pleted, by  multiplying  the  equation  into  four  times  the  co- 
efficient of  the  highest  power  of  the  unknown  quantity  and 
adding  to  both  sides,  the  square  of  the  co-efficient  of  the  lowest 
power. 

The  advantage  of  this  method  is,  that  it  avoids  the  intro- 
duction of  fractions,  in  completing  the  square. 

This  will  be  seen,  by  solving  an  equation  by  both  methods. 

Let  ax2  -\-dx=h, 
Completing  the  square  by  the  rule  just  given ; 

4a2x2  +4adx+d2  =4ah+d2, 
Extracting  the  root,         2ax+d=±:V4ah-{-d2 

A     ,                                                —  d±:^4ah+d2 
And  x= 

2a 

Completing  the  square  of  the  given  equation  by  Arts.  322 

dx      d2       h       d-2 
and  327;  z2  +  —  +  —  =  -  +  — 

a      4a2      a     4a2 


Extracting  the  root,  x+  — -  =  dz  V/  -  a 

2«  v    a     4a2 


And 


d  _l_  *  A       d2 

x=  —  — -± V  --\ 

2a       v    a     4a2 


If  a=l,  the  rule  will  be  reduced  to  this:  "Multiply  the 
equation  by  4,  and  add  to  both  sides  the  square  of  the  co- 
efficient of  x" 

Let    x2+dx=hf  • 

Completing  the  square,      4x2  -\-4dx-\-d2  =4h+d2 

Extracting  the  root,  2x-\-d=ziz^4h+d2 

-d±zV4h+d2 
And  x= - 

\.  Reduce  the  equation  Zx2 -\-5x~42 

Completing  the  square,  S6x2  +60#+25~529 

Therefore,  x=3. 
16* 


180  ClUADRATIC      EaUATIONS. 

2.  Reduce  the  equation  x2  —  15a:—  —  54 

Completing  the  square,         4a:2—  60a;+225=9 
Therefore,  2a;=15db3=18   or  12. 

330.  In  the  square  of  a  binomial,  the  first  and  last  terms 
are  always  positive.  For  each  is  the  square  of  one  of  the 
terms  of  the  root.  (Art.  109.)  But  every  square  is  positive. 
(Art.  233.)  If  then  —x2  occurs  in  an  equation,  it  can  not, 
with  this  sign,  form  a  part  of  the  square  of  a  binomial.  But 
if  all  the  signs  in  the  equation  be  changed,  the  equality  of 
the  sides  will  be  preserved,  (Art.  182,)  the  term  —x2  will 
become  positive,  and  the  square  may  be  completed. 

1.  Reduce  the  equation  —  x2  Jt2x=d—  h 

Changing  all  the  signs,  x2—2x=h  —  d 


Therefore,  x=l±>/l+h—d. 

2.  Reduce  the  equation  4x—x2  =  — 12. 

Ans.  a:=2dbv/16. 

331.  In  a  quadratic  equation,  the  first  term  x2  is  the 
square  of  a  single  letter.  But  a  binomial  quantity  may  con- 
sist of  terms,  one  or  both  of  which  are  already  powers. 

Thus  x3+a  is  a  binomial,  and  its  square  is 

x6+2ax3-\-a2, 

where  the  index  of  x  in  the  first  term  is  twice  as  great  as  in 
the  second.  When  the  third  term  is  deficient,  the  square 
may  be  completed  in  the  same  manner  s,s  that  of  any  other 
binomial.  For  the  middle  term  is  twice  the  product  of  the 
roots  of  the  two  others. 

So  the  square  of  xn+a,  is  x2  n  +2axn  +  a2 . 

And  the  square  of  xn+a,  is  xn+2axn  +a2 . 
Therefore, 

333.  Any  equation  which  contains  only  two  different 
powers  or  roots  of  the  unknown  quantity,  the  index  of  one 
of  which  is  twice  that  of  the  other,  may  be  resolved  in  the 
same  manner  as  a  quadratic  equation,  by  completing  the 
square. 


aUADRATIC      EQUATIONS.  187 

It  must  be  observed,  however,  that  in  the  binomial  root, 
the  letter  expressing  the  unknown  quantity  may  still  have  a 
fractional  or  integral  index,  so  that  a  farther  extraction, 
according  to  Art.  314,  may  be  necessary. 

1.  Reduce  the  equation  x*—  x2=b— a 

Completing  the  square,  x*  —x2  +  \— J +&— a 

Extracting  and  transposing,       x2  =^±  y/^+b—a 


Extracting  again,  (Art.  314,)     ar=dc  ^£=fcV(i+&— a). 

2.  Reduce  the  equation  x2n— 4bxn=a 

Answer     x= zb  v/26=b%/(462-f  a). 

3.  Reduce  the  equation  x-\-4s/x=h— n 
Completing  the  square,  x-\-4^/x-{-4t=h  —  n+4: 
Extracting  and  transposing,  ^/x——2±\^h  —  n+4 
Involving,  x=(— 2±Vh— n+4)2 

4.  Reduce  the  equation  x»+8xn=a+b 

l         1 
Completing  the  square,  x*-\-8zn-\-l6=a-\-b-{-16 


Extracting  and  transposing,     xn  =  —  idz^a  +  b-\-\6 
Involving,  x=(— 4±\/a+b+16)n. 

333.  The  solution  of  a  quadratic  equation,  whether  pure 
or  affected,  gives  two  results.  For  after  the  equation  is  re- 
duced, it  contains  an  ambiguous  root.  In  a  pure  quadratic, 
this  root  is  the  whole  value  of  the  unknown  quantity. 

Thus  the  equation  x2=64 

Becomes,  when  reduced,  x=dz\/64. 

That  is,  the  value  of  x  is  either  +8  or  —8,  for  each  of 
these  is  a  root  of  64.  Here  both  the  values  of  x  are  the 
same,  except  that  they  have  contrary  signs.  This  will  be 
the  case  in  every  pure  quadratic  equation,  because  the  whole 
of  the  second  member  is  under  the  radical  sign.  The  two 
values  of  the  unknown  quantity  will  be  alike,  except  that 
one  will  be  positive,  and  the  other  negative. 

334.  But  in  affected  quadratics,  a  part  only  of  one  side 
of  the  reduced  equation  is  under  the  radical  sign.     When 


188  aUADRATIC      EaUATIONS. 

this  part  is  added  to,  or  subtracted  from,  that  which  is  with- 
out the  radical  sign ;  the  two  results  will  differ  in  quantity, 
and  will  have  their  signs  in  some  cases  alike,  and  in  others 
unlike. 

1.  The  equation  :c2+8:c=20 
Becomes  when  reduced,           x=~- 4ztV  16+20 
That  is,  x=  —  4d=6. 

Here  the  first  value  of  x  is,  —4+6= +2  )  one  "positive,  and 
And  the  second  is,  —4—6=  — 10  )  the  other  negative. 

2.  The  equation  x2  —  8x=  — 15 
Becomes  when  reduced,  '       x=4db  ^16—15 
That  is,  x=4±l. 

Here  the  first  value  of  x  is,  4+1  =  +5  )  ,     , 
And  the  second  is,  4  —  1  =  +3  )  " 

That  these  two  values  of  x  are  correctly  found,  may  be 
proved,  by  substituting  first  one  and  then  the  other,  for  x 
itself,  in  the  original  equation.     (Art.  197.) 

Thus  52-8X5=25-40=-15 
And    a2-8X3=9-24=-15. 

335.  The  value  of  the  unknown  quantity,  in  an  equa- 
tion, is  called,  by  mathematicians,  a  root  of  the  equation. 
The  term  root  has  here  a  meaning,  different  from  that  given  it 
in  the  section  on  radical  quantities.  The  root  of  a  quantity 
is  a  factor  which  multiplied  into  itself  will  produce  that  quan- 
tity. (Art.  252.)  But  a  root  of  an  equation  is  the  value  of 
the  unknown  quantity.  It  is  that  which  substituted  for  the 
unknown  quantity  will  answer  the  conditions  of  the  equation. 
In  a  pure  quadratic,  the  two  meanings  of  the  term  root  coin- 
cide ;  for  when  the  equation  is  reduced,  the  whole  value  of 
the  unknown  quantity  is  under  the  radical  sign.  This  root 
is  also  a  root  of  the  equation.  But  in  affected  quadratics,  a 
part  only  of  one  side  of  the  reduced  equation  is  under  the 
radical  sign.  (Art.  331.)  The  root  indicated  by  this  sign, 
therefore,  is  only  a  part  of  the  root  of  the  equation  ;  a  part 
of  the  value  of  the  unknown  quantity.  In  the  last  example 
above,  it  is  V 16— 15=1;  but  4  +  1  is  a  root  of  the  equation* 


QUADRATIC      EQUATIONS.  189 

336.  In  the  reduction  of  an  affected  quadratic  equation, 
the  value  of  the  unknown  quantity  is  frequently  found  to  be 
imaginary. 

Thus  the  equation  x2  —  8x=  —  20 


Becomes,  when  reduced,  #=4zh  >/i6— 20 

That  is,  x—±±:\/~^i. 

Here  the  root  of  the  negative  quantity  —4  can  not  be 
assigned,  (Art.  274,)  and  therefore  the  value  of  x  can  not  be 
found.  There  will  be  the  same  impossibility,  in  every  in- 
stance in  which  the  negative  part  of  the  quantities  under  the 
radical  sign  is  greater  than  the  positive  part. 

337*  Whenever  one  of  the  values  of  the  unknown  quan- 
tity, in  a  quadratic  equation,  is  imaginary,  the  other  is  so 
also.     For  both  are  equally  affected  by  the  imaginary  root. 

Thus  in  the  example  above, 

The  first  value  of  x  is,  4+  \/— 4, 

And  the  second  is,  4—  v/  —  4;  each  of  which 
contains  the  imaginary  quantity  \/— -4. 

338.  An  equation  which  when  reduced  contains  an  im- 
aginary root,  is  often  of  use,  to  enable  us  to  determine 
whether  a  proposed  question  admits  of  an  answer,  or  in- 
volves an  absurdity. 

Suppose  it  is  required  to  divide  8  into  two  such  parts,  that 
the  product  will  be  20. 

If  x  is  one  of  the  parts,  the  other  will  be  8—x,  (Art.  199.) 
By  the  conditions  proposed,  •       (8— x)Xx=20 

This  becomes,  when  reduced,  x=4±zy/—4. 

Here  the  imaginary  expression  y/  —  4  shows  that  an  an- 
swer is  impossible ;  and  that  there  is  an  absurdity  in  suppos- 
ing that  8  may  be  divided  into  two  such  parts,  that  their 
product  shall  be  20. 

339.  Although  a  quadratic  equation  has  two  solutions, 
yet  both  these  may  not  always  be  applicable  to  the  subject 
proposed.  The  quantity  under  the  radical  sign  may  be  pro- 
duced either  from  a  positive  or  a  negative  root.     But  both 


190  QUADRATIC      EQUATIONS. 

these  roots  may  not,  in  every  instance,  belong  to  the  problem 
to  be  solved.     See  Art.  316. 

Divide  the  number  30  into  two  such  parts,  that  their  pro- 
duct may  be  equal  to  8  times  their  difference. 

If  x=  the  lesser  part,  then  30— x=  the  greater. 

By  the  supposition,  ^X(30-x)  =  8X(30— 2x). 

This  reduced,  gives  #=23dzl7=40  or  6=  the  lesser  part. 

But  as  40  can  not  be  a  part  of  30,  the  problem  can  have 
but  one  real  solution,  making  the  lesser  part  6,  and  the  greater 
part  24. 

Quadratic  Equations  containing  two  or  more  Unknown 
Quantities. 

8  10.  There  is  no  general  rule  by  which  all  quadratic 
equations  of  two  or  more  unknown  quantities  can  be  re- 
solved. When  one  of  the  equations  containing  two  unknown 
quantities  is  a  simple  equation,  the  value  of  one  of  the  un- 
known quantities  may  be  found  in  this,  and  substituted  for 
that  quantity,  in  the  other  equation.  This  will  reduce  the 
latter  to  a  quadratic  with  only  one  unknown  quantity,  which 
may  be  resolved  by  the  ordinary  rules.  More  expeditious 
methods  may,  in  some  cases,  be  adopted ;  particularly  when 
the  sum  or  difference  of  two  quantities  is  given  in  one  equa- 
tion, and  the  sum  or  difference  of  their  squares  in  another. 

Ex.  1.  Let     2x+y=  27 
And       3^=210 
From  the  1st  equation    #=|-(27  — y) 

Let  this  value  be  substituted  for  x  in  the  second  equation. 
We  then  have         3yXh(27-y)=2lO, 

Hence,  y2—  27y=  — 140,    an    affected   quad- 

ratic with  only  one  unknown  quantity,  which  when  reduced 
gives  y=20,  and  x=3\. 

2.  There  is  a  certain  number  consisting  of  two  digits. 
The  left-hand  digit  is  equal  to  3  times  the  right-hand  digit ; 
and  if  twelve  be  subtracted  from  the  number  itself,  the  re- 
mainder will  be  equal  to  the  square  of  the  left-hand  digit. 
What  is  the  number? 


QUADRATIC      EQUATIONS.  191 

Let  x=  the  left-hand  digit,  and  y=  the  right-hand  digit. 

As  the  local  value  of  figures  increases  in  a  ten-fold  ratio 
from  right  to  left ;  the  number  required  =  10x+y 

By  the  conditions  of  the  problem,  x=Sy  ) 

And  10x+y—l2=x2  ) 

The  required  number  is,  93. 

3.  To  find  one  of  two  quantities, 
Whose  sum  is  equal  to  h  ;   and 
The  difference  of  whose  squares  is  equal  to  d. 

Let  x=  the  greater  quantity ;    And  y=  the  less. 

1.  By  the  first  condition,  x-\-y=h      ) 

2.  By  the  second,  x2—y2=d  ) 

3.  Transposing  y2  in  the  2d  equation,  x2=d+y2 

4.  By  evolution,  (Art.  314,)  x=  Vd+y2 

5.  Transposing  y  in  the  first  equation,  x—h—y 

6.  Making  the  4th  and  5th  equal,  \^d+y2=h—y 

h2—d 

7.  Therefore,  V= ; — 

y         2h 

4.  To  find  two  numbers  such,  that 

The  product  of  their  sum  and  difference  shall  be  5,  and 
The  product  of  the  sum  of  their  squares  and  the  differ- 
ence of  their  squares  shall  be  65. 

Let  x—  the  greater;         And  y=  the  less. 

1.  By  the  first  condition,  (x+y)x(x— y)  =  5  ) 

2.  By  the  second,  (x2+y2)x(x2—  y2)=65  ) 

3.  Multiply  the  factors  in  the  1st,  (Art.  Ill,)  x2  —  y2=5 

4.  Dividing  the  2d  by  the  3d,  (Art.  1 14.)         x2+y2  =  lS 

5.  Adding  the  3d  and  4th,  2x2  =  18 

6.  Therefore,  x=S,  the  greater  number, 

7.  And  y=2,  the  less. 

In  the  4th  step,  the  first  member  of  the  second  equation  is 
divided  by  x2  —  y2,  and  the  second  member  by  5,  which  is 
equal  to  x2—y2. 


192  aUADRATIC      EdUATIONS. 

5.  To  find  two  numbers  whose  difference  is  8,  and  product 
240. 

6.  To  find  two  numbers, 

Whose  difference  shall  be  12,  and 
The  sum  of  their  squares  1424. 

Let  x=  the  greater;         And  y=  the  less. 

1.  By  the  first  condition,  x— y=12 

2.  By  the  second,  #2+y2  =  1424 

3.  Transposing  y  in  the  first,         x=y-\-\2 

4.  Squaring  both  sides,  x2=y2  +24y+144 

5.  Transposing  y2  in  the  second,  x2  =  \424—y2 

6.  Making  the  4th  and  5th  equal,  y2  +24?/  + 144  =  1424— y2 

7.  Therefore,  t/=-6d=v/(676)  =  -6±26 

8.  And  #=3/+12=20  +  12=32. 

Properties  of  Quadratic  Equations. 

341.  The  several  terms  of  an  affected  quadratic  equa- 
tion may  be  reduced  to  three;  one  containing  the  square  of 
the  unknown  quantity,  another,  the  quantity  itself  with  its 
co-efficient,  and  the  third,  the  sum  of  the  known  quantities, 
which  may  be  expressed  by  a  single  letter.  Every  such 
equation,  therefore,  may  be  reduced  to  one  of  the  four  fol- 
lowing forms,  differing  in  the  signs  only. 

1.  x2-\-ax—b     ^ 

2.  Xs—  ax=b       [These,   when    re- 

3.  x*  +ax=  —  b 

4.  x2  —  ax=  —  b> 


1.  x=-la±V\a2+b 

2.  x=     %a±^±a2+b 
duped,  become,  ~"|  3>  x=—$a±y/\aa-b 

U.  x=     \a±^\a2-b 

In  the  first  two  of  these  forms,  the  roots  are  never  imagin- 
ary ;  for  the  terms  under  the  radical  sign  are  both  positive. 
But  in  the  third  and  fourth  forms,  whenever  b  is  greater 
than  \a2,  the  expression  \a2  —  b  is  negative,  and  therefore 
its  root  is  impossible. 

In  each  of  the  first  two  forms,  as  V '  \a2  +b  is  greater  than 
\a,  the  square  root  of  \a2 ,  one  value  of  x  is  positive,  the 
other  negative.     In  the  third  form,  as   ^\a2—  b,  when  not 


QUADRATIC      EQUATIONS.  193 

imaginary,  is  less  than  \a,  both  values  of  x  are  negative ; 
and  in  the  fourth  form,  both  are  positive. 

342.  In  the  equation  x2+ax=b,  or  x2+ax  —  b=0,  the 

C-ia+V±a*+b 
two  values  ot  x  are  <      „  .- 


Therefore,  transposing,  <        ,  .- 

The  first  of  these  is  the  difference,  and  the  other  is  the 
sum  of  the  two  expressions  x  +  l2a  and   \/^a2-\-b. 

This  sum  and  difference  multiplied  together  (Art.  Ill,)  give 

(x+%a)2-(la2+b)  =  0. 
Reducing  this,  we  have 

x2+av— b=0,  the  original  equation. 

We  have  come  to  this  result,  by  multiplying  x  minus  the 
first  value  of  x,  into  x  minus  the  second  value  of  x. 


Let  v  ——\a-\-\/\a2Jrb\  the  first  value  of  x. 

And  v'=—  \a—*S\a2-\-b,   the  second  value,  or  root  of  the 

equation, 
We  have  then  x2-\-ax  —  b=(x— v)X(x— v'), 

And  we  obtain  this  general  proposition;  When  all  the 
terms  of  a  quadratic  equation  are  brought  on  one  side,  this 
member  is  the  product  of  two  binomial  factors,  each  consist- 
ing of  the  unknown  quantity  and  one  of  its*  values,  with  a 
contrary  sign. 

Thus  the  roots  of  the  equation  x2  +6^-55=0  are  5  and  -11. 
And    (x+ll)X(x-5)=x2+6x-55. 

Of  consequence,  if  the  first  member  of  the  equation  be 
divided  by  one  of  these  factors,  the  quotient  will  be  the  other 
factor.  When,  therefore,  one  of  the  values  of  the  unknown 
quantity  is  found,  the  other  value  may  be  obtained  by  divid- 
ing the  equation  by  the  binomial  factor  of  which  the  value 
already  found  is  a  part. 

17 


191  QUADRATIC      EQUATIONS. 

Ex.  1.  What  is  the  equation  the  roots  of  which  are  3 
and  — -§? 

2.  If  one  of  the  roots  of  the  equation  x2  —  15a;=-54  is  9, 
what  is  the  other  root  ? 

343.  Of  the  equation  x2+ax=b,  the  two  roots  are 


-ia+>/iaa+b 
—ia—Via2+b. 


When  these  are  added  together,  the  radical  parts  having 
contrary  signs,  disappear;  and  the  sum  of  the  roots  is  —a, 
which,  with  the  contrary  sign,  is  the  co-efficient  of  x,  in  the 
second  term  of  the  equation.     Hence, 

The  sum  of  the  roots  of  a  quadratic  equation  of  three 
terms,  the  sign  being  changed,  is  equal  to  the  co-efficient  of 
the  second  term  of  the  equation. 

344.  If  the  two  roots  above  be  multiplied  together,  the 
result,  being  the  product  of  the  sum  and  difference  of  twTo 
quantities,  (Art.  Ill,)  is  \a2  —  {\a2  -\-b),  that  is,  —b,  which, 
with  the  contrary  sign,  is  the  second  member  of  the  equa- 
tion  x2  -{-ax=b.     Hence, 

The  second  member  of  a  quadratic  equation  reduced  to 
three  terms,  the  signs  being  changed,  is  equal  to  the  product 
of  the  two  roots  of  the  equation. 

Examples  of  Quadratic  Equations, 

1.  Reduce  3x2  —  9x— 4=80.  Ans.  x=7,  or  —4. 

_>  36 — x  , 

2.  Reduce  Ax* =46.  Ans.  x=12,  or  — £. 

x  * 

14— x 

3.  Reduce  4x —-  =  14.  Ans.  x=4,  or  —  J. 

x+1 

_    .  Sx— 3  Sx  —  6 

4.  Reduce  5x — =2x-\ —     Ans.  x=4,  or—  1. 

x— 3  2 

„    ,  16      100-9z 

5.  Reduce — — — =3.  Ans.  x=4,  or  2TV 

x  4x2  3 

6.  Reduce  - — -  +  1  =  10—  — —         Ans.  x=12,  or  6. 

x—4  2 


QUADRATIC      EQUATIONS.  195 

^    B         ar+4     7—x     4x+7  .  • 

7.  Reduce  — r=  -5 1.      Ans.  #=21,  or  5. 

#3  _  10iC2  +1 

8.  Reduce  — n^*"""3-      Ans-  *'pl>  or  ~28' 

xa—  Gx  +9 

9.  Reduce  — rr  +  -==3-  Ans-  x==2' 

2x       x 1 

10.  Reduce  — — — =x-9.  Ans.  x=10. 

x+2         6 


11.  Reduce  -  +  -  = 
a     x     a 


12. 


x     a     2  A  .  ,     ,- 

-  +  -  =  —  Ans.  x=l±vl-a8. 

a     z      a 

Reduce  x*+ax2=b.     Ans.  a;=(-?±VH-)- 

\     2  4  / 

<j»  6  o*  3 

13.  Reduce  —  -  —  =  -  — •  Ans.  x=Vi- 

14.  Reduce  2x^+2x^=2.  Ans.  x=f 

15.  Reduce  i:c-iv'#=22±.  Ans.  #=49. 

16.  Reduce  2x*—x*  +96=99.  Ans.  x=^^6. 

17.  Reduce  (10+.t)^-(10+x)¥=2.  Ans.  x=6. 

18.  Reduce  3#2n— 2#*=8.  Ans.  x=y2. 

19.  Reduce  2(1+3— a*)  —  \/l+#-#2  =  -f 

Ans.  x=±+}</41. 

20.  Reduce   ^c3  — a2=x— b.     Ans.  #=-=h\/ 

2  126 

-,    '  v/4^+2      4—  s/x 

21.  Reduce  — - — —  = — Ans.  x=4. 

4+^/x  s/x 


22.  Reduce  #5+#5=756.  Ans.  #=243. 

, 21 

23.  Reduce    v2#+l+2v/#=-7=-  Ans.  #=4. 

\/2x  +  l 

24.  Reduce  2>/x—a+3y/2x= ■  Ans.  #=9a. 

25.  Reduce  #+16-7 ^#+16=10-4 ^#  +  16.    Ans.  #=9. 


196  QUADRATIC      EQUATIONS. 

26.  Reduce   <Jx5  +  y/x3=Gx/x. 

Dividing  by   y/x,         x2-\-x=G.  Ans.  x=2. 

^    .         4x-5      Sx-7      to+23 

.    27.  Keduce -— — -  =  — — Ans.  x=2. 

x  3x+7         VSx 

3  6  11 

28.  Keduce  — -f    0  ,  ^    —  —  •  Ans.  a?=3. 

%       3  3 

29.  Reduce  (x— 5)   —  3(#-5)2=40.  Ans.  x=9. 

30.  Reduce  #+v/x-r-6=2+3  Vx+6.  Ans.  #=10. 

Problems  producing  Quadratic  Equations. 

Prob.  1.  A  merchant  has  a  piece  of  cotton  cloth,  and  a 
piece  of  silk.  The  number  of  yards  in  both  is  110:  and  if 
the  square  of  the  number  of  yards  of  silk  be  subtracted  from 
80  times  the  number  of  yards  of  cotton,  the  difference  will 
be  400.     How  many  yards  are  there  in  each  piece  ? 

Let  x=  the  yards  of  silk. 

Then     110-     a? —    tho  yetrdo  of   COtton. 

By  supposition,         400=80  X  (llO-x)-x3 
Therefore,  x=  -40zi=  V 10000=  -40=bl00. 

The  first  value  of  x,  is  —40  +  100=60,  the  yards  of  silk; 

And  110— x=110— 60=50,  the  yards  of  cotton. 

The  second  value  of  x,  is  —40— 100=  — 140 ;  but  as  this 
is  a  negative  quantity,  it  is  not  applicable  to  goods  which  a 
man  has  in  his  possession. 

Prob.  2.  The  ages  of  two  brothers  are  such,  that  their 
sum  is  45  years,  and  their  product  500.  What  is  the  age  of 
each  ?  Ans.  25  and  20  years. 

Prob.  3.  To  find  two  numbers  such,  that  their  difference 
shall  be  4,  and  their  product  117. 

Let  x=  one  number,  and  x +4=  the  other. 
By  the  conditions,  (x-\-4)Xx=117 

This  reduced,  gives,  x=—  2^=^121  =  —  2=hll. 

One  of  the  numbers  therefore  is  9,  and  the  other  13. 


QUADRATIC      EQUATIONS.  197 

Prob.  4.  A  merchant  having  sold  a  piece  of  cloth  which 
cost  him  30  dollars,  found  that  if  the  price  for  which  he  sold 
it  were  multiplied  by  his  gain,  the  product  would  be  equal  to 
the  cube  of  his  gain.     What  was  his  gain  ? 

Let  x=  the  gain. 

Then  30+a;=  the  price  for  which  the  cloth  was  sold 
By  the  statement,  x3  =  (30+x*)Xx 

Therefore,  »=Jdb  v^+ 30=|zbV. 

The  first  value  of  x  is  £+V=+6-  ) 

The  second  value  is       ^— 121  =  —  5.  S 

As  the  last  answer  is  negative,  it  is  to  be  rejected  as  incon- 
sistent with  the  nature  of  the  problem,  (Art.  339,)  for  gain 
must  be  considered  positive. 

Prob.  5.  To  find  two  numbers  whose  difference  shall  be  3, 
and  the  difference  of  their  cubes  117. 

Let  x=  the  less  number. 
Then  x+3~  the  greater. 

By  supposition,  (x+3)3-  — a;3  =  117 

Expanding  (x  +  3)3  (Art.  232,)  9x2  +27.z=  117-27=90 
And  ^=-fzbv/^-=-|±J. 

The  two  numbers,  therefore,  are  2  and  5. 

Prob.  6.  To  find  two  numbers  whose  difference  shall  be 
12,  and  the  sum  of  their  squares  1424.  % 

Ans.    The  numbers  are  20  and  32. 

Prob.  7.  Two  persons  draw  prizes  in  a  lottery,  the  differ- 
ence of  which  is  120  dollars,  and  the  greater  is  to  the  less, 
as  the  less  to  10.     What  are  the  prizes? 

Ans.  40  and  160. 

Prob.  8.  What  two  numbers  are  those  whose  sum  is  6, 
and  the  sum  of  their  cubes  72  ?  Ans.  2  and  4. 

Prob.  9.  Divide  the  number  56  into  two  such  parts,  that 
their  product  shall  be  640. 

Putting  x  for  one  of  the  parts,  we  have,  #=28=izl2=40 
or  16. 

17* 


198  QUADRATIC      EQUATIONS. 

In  this  case,  the  two  values  of  the  unknown  quantity  are 
the  two  parts  into  which  the  given  number  was  required  to 
be  divided. 

Prob.  10.  A  gentleman  bought  a  number  of  pieces  of  cloth 
for  675  dollars,  which  he  sold  again  at  48  dollars  by  the  piece, 
and  gained  by  the  bargain  as  much  as  one  piece  cost  him. 
What  was  the  number  of  pieces  ?  Ans.   15. 

Prob.  11.  A  and  B  started  together,  for  a  place  150  miles 
distant.  A's  hourly  progress  was  3  miles  more  than  2?'s,  and 
he  arrived  at  his  journey's  end  8  hours  and  20  minutes  before 
B.     What  was  the  hourly  progress  of  each  ? 

Ans.  9  and  6  miles. 
Prob.  12.  The  difference  of  two  numbers  is  6 ;  and  if  47 
be  added  to  twice  the  square  of  the  less,  it  will  be  equal  to 
the  square  of  the  greater.     What  are  the  numbers  ? 

Ans.   17  and  11. 
Prob.  13.   A  and  B  distributed  1200  dollars  each,  among 
a  certain  number  of  persons.     A  relieved  40  persons  more 
than  B,  and  B  gave  to  each  individual  5  dollars  more  than 
A.     How  many  were  relieved  by.  A  and  B  ? 

Ans.  120  by  A,  and  80  by  B. 

Prob.  14.  Find  two  numbers  whose  sum  is  10,  and  the  sum 
of  their  squares  58.  Ans.  7  and  3. 

Prob.  15.  Several  gentlemen  made  a  purchase  in  company 
for  175  dollars.  Two  of  them  having  withdrawn,  the  bill 
was  paid  by  the  others,  each  furnishing  10  dollars  more  than 
would  have  been  his  equal  share  if  the  bill  had  been  paid  by 
the  whole  company.  What  was  the  number  in  the  company 
at  first?  Ans.   7.  * 

Prob.  16.  A  merchant  bought  several  yards  of  linen  for 
60  dollars,  out  of  which  he  reserved  15  yards,  and  sold  the 
remainder  for  54  dollars,  gaining  10  cents  a  yard.  How 
many  yards  did  he  buy,  and  at  what  price  ? 

Ans.  75  yards,  at  80  cents  a  yard. 

Prob.  17.  A  and  B  set  out  from  two  towns,  which  were 
247  miles  distant,  and  travelled  the  direct  road  till  they  met. 
A  went  9  miles  a  day ;  and  the  number  of  days  which  they 
travelled  before  meeting,  was  greater  by  3,  than  the  Dumber 
of  miles  which  B  went  in  a  day.  How  many  miles  did  e; 
travel?  Ans.  A  went  117,  and  B  130  miles. 


QUADRATIC      EQUATIONS.  199 

Prob.  18.  A  gentleman  bought  two  pieces  of  cloth,  the 
finer  of  which  cost  4  shillings  a  yard  more  than  the  other. 
The  finer  piece  cost  £18  ;  but  the  coarser  one,  which  was  2 
yards  longer  than  the  finer,  cost  only  £16.  How  many 
yards,  were  there  in  each  piece,  and  what  was  the  price  of  a  . 
yard  of  each  ? 

Ans.  There  were  18  yards  of  the  finer  piece,  and  20  of 
the  coarser;  and  the  prices  were  20  and  16  shillings. 

Prob.  19.  A  merchant  bought  54  gallons  of  Madeira  wine, 
and  a  certain  quantity  of  Teneriffe.  For  the  former,  he  gave 
half  as  many  shillings  by  the  gallon,  as  there  were  gallons  of 
Teneriffe,  and  for  the  latter,  4  shillings  less  by  the  gallon. 
He  sold  the  mixture  at  10  shillings  by  the  gallon,  and  lost 
£28  16s.  by  his  bargain.  Required  the  price  of  the  Madeira, 
and  the  number  of  gallons  of  Teneriffe. 

Ans.  The  Madeira  cost  18  shillings  a  gallon,  and  there 
were  36  gallons  of  Teneriffe. 

Prob.  20.  If  the  square  of  a  certain  number  be  taken  from 
40,  and  the  square  root  of  this  difference  be  increased  by  10, 
and  the  sum  be  multiplied  by  2,  and  the  product  divided  by 
the  number  itself,  the  quotient  will  be  4.  What  is*  the 
number?  A  Ans.  6. 

Prob.  21.  A  person  being  asked  his  age,  replied,  If  you  add 
the  square  root  of  it  to  half  of  it,  and  subtract  12,  the  remain- 
der will  be  nothing.     What  was  his  age?     Ans.   16  years. 

Prob.  22.  Two  casks  of  wine  were  purchased  for  58  dol- 
lars, one  of  which  contained  5  gallons  more  than  the  other, 
and  the  price  by  the  gallon,  was  2  dollars  less  than  \  of  the 
number  of  gallons  in  the  smaller  cask.  Required  the  num- 
ber of  gallons  in  each,  and  the  price  by  the  gallon. 

Ans.  The  numbers  were  12  and  17,  and  the  price  by  the 
gallon  2  dollars. 

Prob.  23.  In  a  parcel  which  contains  24  coins  of  silver 
and  copper,  each  silver  coin  is  worth  as  many  cents  as  there 
are  copper  coins,  and  each  copper  coin  is  worth  as  many 
cents  as  there  are  silver  coins ;  and  the  whole  are  worth  2 
dollars  and  16  cents.     How  many  are  there  of  each? 

Ans.  6  of  one,  arid  18  of  the  other. 

Prob.  24.  A  person  bought  a  certain  number  of  oxen  for 
80  guineas.  If  he  had  received  4  more  oxen  for  the  same 
money,  he  would  have  paid  one  guinea  less  for  each.  What 
was  the  number  of  oxen  ?  Ans.   16. 


200  QUADRATIC      EQUATIONS. 

Prob.  25.  There  are  two  numbers  such,  that  if  the  less  be 
taken  from  three  times  the  greater,  the  remainder  will  be  35; 
and  if  four  times  the  greater  be  divided  by  three  times  the 
less  +1,  the  quotient  will  be  equal  to  the  less.  What  are 
the  numbers  ? 

Prob.  26.  What  two  numbers  are  those,  whose  difference, 
sum,  and  product  are  as  the  numbers  2,  3,  and  5  ? 

Ans.  10  and  2. 

Prob.  27.  Given    (  U4x+2!/)=  6  >  To  find  the  values  of 

And       I      5xy        =50  $      x  and  y. 

Ans.  y =5,  or  4,  x =2,  or  2  J. 

„    _  ~.  C  x  +v  —a  ?  To  find  the  values  of  x 

Prob.  28.  Given    }     „  ,     „     ,  £  . 

(x2+y2  =  b  >      and  y. 

Substitution. 

34t>.  In  the  reduction  of  Quadratic  Equations,  as  well 
as  in  other  parts  of  Algebra,  a  complicated  process  may  be 
rendered  much  more  simple,  by  introducing  a  new  letter 
which  shall  be  made  to  represent  several  others.  This  is 
termed  substitution.  A  letter  may  be  put  for  a  compound 
quantity  as  well  as  for  a  single  number.     Thus  in  the  equation 

x2  -2ax=%+  ^86-64 -f- A, 

we  may  substitute  b,  for     f  -f-v/86  —  64+ A.      The  equation 

will  then  become  x2—  2ax=b,     and  when  reduced 

will  be  x=a±^/a2-\-b. 

After  the  operation  is  completed,  the  compound  quantity 
for  which  a  single  letter  has  been  substituted,  may  be  restored* 
The  last  equation,  by  restoring  the  value  of  b,  will  become 


Reduce  the  equation         ax— 2x— d=bx— x2  —  x 
Transposing,  &c.  x2  +  (a— b—  \)Xx=d 

Substituting  h  for  (a— 6—1),  x2+hx=d 

Therefore,  x=  —  -  zb  S/  —  +rf 


Restoring  the  value  of  h,  x——  - d=  \/  i? : — \- <£ 

2  4 


RATIO      AND      PROPORTION.  201 


SECTIO 


RATIO    AND    PROPO 


Art.  346.  The  design  of  mathematical  investigations,  is 
to  arrive  at  the  knowledge  of  particular  quantities,  by  com- 
paring them  with  other  quantities,  either  equal  to,  or  greater 
or  less  than  those  which  are  the  objects  of  inquiry.  The  end 
is  most  commonly  attained  by  means  of  a  series  of  equations 
and  proportions.  When  we  make  use  of  equations,  we 
determine  the  quantity  sought,  by  discovering  its  equality 
with  some  other  quantity  or  quantities  already  known. 

We  have  frequent  occasion,  however,  to  compare  the  un- 
known quantity  with  others  which  are  not  equal  to  it,  but 
either  greater  or  less.  Here  a  different  mode  of  proceeding 
becomes  necessary.  We  may  inquire,  either  huw  -much  one 
of  the  quantities  is  greater  than  the  other;  or  how  many  times 
the  one  contains  the  other.  In  finding  the  answer  to  either 
of  these  inquiries,  we  discover  what  is  termed  a  ratio  of  the 
two  quantities.  One  is  called  arithmetical  and  the  other  geo- 
metrical ratio.  It  should  be  observed,  however,  that  both 
these  terms  have  been  adopted  arbitrarily,  merely  for  distinc- 
tion's sake.  Arithmetical  ratio,  and  geometrical  ratio  are 
both  of  them  applicable  to  arithmetic,  and  both  to  geometry. 

As  the  whole  of  the  extensive  and  important  subject  of 
proportion  depends  upon  ratios,  it  is  necessary  that  these 
should  be  clearly  and  fully  understood. 

3417.  Arithmetical  ratio  is  the  difference  between  two 
quantities  or  sets  of  quantities.  The  quantities  themselves  are 
called  the  terms  of  the  ratio,  that  is,  the  terms  between  which 
the  ratio  exists.  Thus  2  is  the  arithmetical  ratio  of  5  to  3. 
This  is  sometimes  expressed,  by  placing  two  points  between 
the  quantities  thus,  5.  .3,  which  is  the  same  as  5-3.  Indeed 
the  term  arithmetical  ratio,  and  its  notation  by  points,  are 
almost  needless.  For  the  one  is  only  a  substitute  for  the 
word  difference,  and  the  other  for  the  sign  — . 


202  RATIO      AND      PROPORTION. 

348,  If  both  the  terms  of  an  arithmetical  ratio  be  multi- 
plied or  divided  by  the  same  quantity,  the  ratio  will,  in  effect, 
be  multiplied  or  divided  by  that  quantity. 

Thus  if  a  —  b=r 

Then  multiply  both  sides  by  A,  (Ax.  3,)    ha—hb=hr 

a      b      r 
And  dividing  by  A,  (Ax.  4,)  h~~h~h' 

349.  If  the  terms  of  one  arithmetical  ratio  be  added  to, 
or  subtracted  from,  the  corresponding  terms  of  another,  the 
ratio  of  their  sum  or  difference  will  be  equal  to  the  sum  or 
difference  of  the  two  ratios. 


are  the  two  ratios, 


If        a-b 

And    d—h 

Then  (a+d)-(b+h)  =  (a-b)  +  (d-h).  For  each  =a+d-b-h. 
And    (a-d)-(b-h)  =  (a-b)-(d-h).     For  each  =a-d-b+h. 

Thus  the  arithmetical  ratio  of  11.  .4  is  7 

And  the  arithmetical  ratio  of      5 . .  2  is  3 

The  ratio  of  the  sum  of  the  terms   16.  .6  is  10,  the  sum  of 

the  ratios. 
The  ratio  of  the  difference  of  the  terms  6.  .2  is  4,  the  differ- 
ence of  the  ratios. 

350.  Geometrical  ratio  is  that  relation  between  quan- 
tities which  is  expressed  by  the  giuotient  of  the  one  divided 
by  the  other  * 

Thus  the  ratio  of  8  to  4,  is  f  or  2.  For  this  is  the  quo- 
tient of  8  divided  by  4.  In  other  words,  it  shows  how  often 
4  is  contained  in  8. 

In  the  same  manner,  the  ratio  of  any  quantity  to  another 
may  be  expressed  by  dividing  the  former  by  the  latter,  or, 
which  is  the  same  thing,  making  the  former  the  numerator 
of  a  fraction,  and  the  latter  the  denominator. 

a 
Thus  the  ratio  of  a  to  b  is  r« 

o 

7    i     -L 

The  ratio  of  d+h  to  b+c,  is  -j— — 

b+c 

*  See  Note  F. 


RATIO      AND      PROPORTION.  203 

3ol.  Geometrical  ratio  is  also  expressed  by  placing  two 
points,  one  over  the  other,  between  the  quantities  compared. 

Thus  a  \  b  expresses  the  ratio  of  a  to  b ;  and  12:4  the 
ratio  of  12  to  4.  The  two  quantities  together  are  called  a 
couplet,  of  which  the  first  term  is  the  antecedent,  and  the  last, 
the  consequent. 

«S52.  This  notation  by  points,  and  the  other  in  the  form 
of  a  fraction,  may  be  exchanged  the  one  for  the  other,  as 
convenience  may  require  ;  observing  to  make-  the  antecedent 
of  the  couplet,  the  numerator  of  the  fraction,  and  the  con- 
sequent the  denominator. 

b 

Thus  10  I  5  is  the  same  as  -^  and  b  !  d,  the  same  as  -v 

3*>3.  Of  these  three,  the  antecedent,  the  consequent,  and 
the  ratio,  any  two  being  given,  the  other  may  be  found. 

Let  a~  the  antecedent,  c=  the  consequent,  r=  the  ratio. 

By  definition  r—  - ;  that  is,  the  ratio  is  equal  to  the  antece- 
dent divided  by  the  consequent. 

Multiplying  by  c,  a=cr,  that  is,  the  antecedent  is  equal  to 
the  consequent  multiplied  into  the  ratio. 

Dividing  by  r,  c=  ->   that  is,  the  consequent  is  equal  to  the 

antecedent  divided  by  the  ratio. 

Cor.  1.  If  two  couplets  have  their  antecedents  equal,  and 
their  consequents  equal,  their  ratios  must  be  equal.* 

Cor.  2.  If,  in  two  couplets,  the  ratios  are  equal,  and  the 
antecedents  equal,  the  consequents  are  equal ;  and  if  the 
ratios  are  equal  and  the  consequents  equal,  the  antecedents 
are  equal. f  *s 

Sl«5 1 .  If  the  two  quantities  compared  are  equal,  the  ratio 
is  a  unit,  or  a  ratio  of  equality.  The  ratio  of  3X6  I  18  is  a 
unit,  for  the  quotient  of  any  quantity  divided  by  itself  is  1. 

If  the  antecedent  of  a  couplet  is  greater  than  the  conse- 
quent, the  ratio  is  greater  than  a  unit.  For  if  a  dividend  is 
greater  than  its  divisor,  the  quotient  is  greater  than  a  unit. 
Thus  the  ratio  of  18  :  6  is  3.  (Art.  123,  cor.)  This  is  called 
a  ratio  of  greater  inequality. 

*  Euclid  7,  5.  f  Euclid  9,  5. 


204  RATIO      AND      PROPORTION. 

On  the  other  hand,  if  the  antecedent  is  less  than  the  con- 
sequent, the  ratio  is  less  than  a  unit,  and  is  called  a  ratio  of 
less  inequality.  Thus  the  ratio  of  2  I  3,  is  less  than  a  unit, 
because  the  dividend  is  less  than  the  divisor. 

When  one  ratio  of  inequality  is  compared  with  another, 
if  both  are  ratios  of  greater,  or  of  less  inequality,  they  are 
said  to  subsist  in  the  same  sense.  But  if  one  is  a  ratio  of 
greater  inequality,  and  the  other  a  ratio  of  less  inequality, 
they  are  said  to  subsist  in  a  contrary  sense. 

3«>«>.  Inverse  or  reciprocal  ratio  is  the  ratio  of  the 
reciprocals  of  two  quantities.     See  Art.  43. 

Thus  the  reciprocal  ratio  of  6  to  3,  is  }  to  i,  that  is,  |-r^. 
The  direct  ratio  of  a  to  &  is  t>    that  is,   the   antecedent 

divided  by  the  consequent. 

,..11  1       1       1      b      b 

1  he  reciprocal  ratio  is  -  :  T  or  --tt  =  -Xt  =  -: 
a     b         a      b      a     1      a 

that  is  the  consequent  b  divided  by  the  antecedent  a. 

Hence  a  reciprocal  ratio  is  expressed  by  inverting  the  frac- 
tion which  expresses  the  direct  ratio ;  or  when  the  notation 
is  by  points,  by  inverting  the  order  of  the  terms. 

Thus  a  is  to  b,  inversely,  as  b  to  a. 

3%>6.  Compound  ratio  is  the  ratio  of  the  products,  of 
the  corresponding  terms  of  two  or  more  simple  ratios* 

Thus  the  ratio  of  6  :  3,  is  2 

And  the  ratio  of  12  :  4,  is  3 


The  ratio  compounded  of  these  is  72  :  12=6. 

Here  the  compound  ratio  is  obtained  by  multiplying 
together  the  two  antecedents,  and  also  the  two  consequents, 
of  the  simple  ratios. 

So  the  ratio  compounded, 

Of  the  ratio  of  a  \  b 

And  the  ratio  of  c  \  d 

And  the  ratio  of  h  \  y 

Is  the  ratio  of  ach  \  bdy=  -. 


My 


*  See  Note  G. 


RATIO      AND      PROPORTION.  205 

Compound  ratio  is  not  different  in  its  nature  from  any- 
other  ratio.  The  term  is  used,  to  denote  the  origin  of  the 
ratio,  in  particular  cases. 

Cor.  The  compound  ratio  is  equal  to  the  product  of  the 
simple  ratios. 


The  ratio  of  a  \  b,  is 

The  ratio  of  c  :  d ,  is 

The  ratio  of  h  \  y,  is 


a 
b 
c 
d 
h 

y 

ach 


And  the  ratio  compounded  of  these  is   7-7- >   which  is  the 

product  of  the  fractions  expressing  the  simple  ratios.  (Art.  158.) 

3o7.  If,  in  a  series  of  ratios,  the  consequent  of  each  pre- 
ceding couplet,  is  the  antecedent  of  the  following  one,  the 
ratio  of  the  first  antecedent  to  the  last  consequent  is  equal  to 
that  which  is  compounded  of  all  the  intervening  ratios .* 

Thus,  in  the  series  of  ratios  a  \  b 

b  :  c 
c  :  d 

d:h 

the  ratio  of  a  '.  h  is  equal  to  that  which  is  compounded  of  the 
ratios  of  a  I  b,  of  b  '.  c,  of  c  :  d,  of  d  :  h.     For  the  compound 

,1  ■  i     •     aocd     a  7 

ratio  by  the  last  article  is  t~tt  =  t  or  a\h.     (Art.  149.) 

In  the  same  manner,  all  the  quantities  which  are  both 
antecedents  and  consequents  will  disappear  when  the  frac- 
tional product  is  reduced  to  its  lowest  terms,  and  will  leave 
the  compound  ratio  to  be  expressed  by  the  first  antecedent 
and  the  last  consequent. 

358.  A  particular  class  of  compound  ratios  is  produced, 
by  multiplying  a  simple  ratio  into  itself  or  into  another  equal 
ratio.  These  are  termed  duplicate,  triplicate3  quadruplicate, 
&c.  according  to  the  number  of  multiplications. 

*  This  is  the  particular  case  of  compound  rr;  16  which  is  treated  of  in  the 
5th  book  of  Euclid.    See  the  editions  of  Simson  and  Playfair. 

18 


206  RATIO      AND      PROPORTION. 

A  ratio  compounded  of  two  equal  ratios,  that  is,  the  square 
of  the  simple  ratio,  is  called  a  duplicate  ratio. 

One  compounded  of  three,  that  is,  the  cube  of  the  simple 
ratio,  is  called  triplicate,  &c. 

In  a  similar  manner,  the  ratio  of  the  square  roots  of  two 
quantities,  is  called  a  subduplicate  ratio;  that  of  the  cube 
roots  a  subtriplicatei&tlo,  &c.  / 

Thus  the  simple  ratio  of  a  to  b,  is  a  :  b 
The  duplicate  ratio  of  a  to  b,  is  a2  :  62 
The  triplicate  ratio  of  a  to  b,  is  a3  :  63 
The  subduplicate  ratio  of  a  to  b,  is    -/a  I  s/b 
The  subtriplicate  of  a  to  b,  is   $/a  :  %/b,  &c. 

The  terms  duplicate,  triplicate,  &c.  ought  not  to  be  con- 
founded with  double,  triple,  &c.# 

The  ratio  of  6  to  2,  is  6  \  2=3 

Double  this  ratio,  that  is,  £w?ice  the  ratio,  is    12  I  2=6  £ 

Triple  the  ratio,  i.  e.  three  times  the  ratio,  is  18  \  2=9  ) 

But  the  duplicate  ratio,  i.e.  the  square  of  the  ratio,  is  62  :  22  =9  ) 

And  the  triplicate  ratio,  i.e.  the  cw&e  of  the  ratio,  is  6 3  :  2 3  =27  ) 

359.  That  quantities  may  have  a  ratio  to  each  other,  it 
is  necessary  that  they  should  be  so  far  of  the  same  nature,  as 
that  one  can  properly  be  said  to  be  either  equal  to,  or  greater, 
or  less  than  the  other.  A  foot  has  a  ratio  to  an  inch,  for  one 
is  twelve  times  as  great  as  the  other.  But  it  can  not  be  said 
that  an  hour  is  either  shorter  or  longer  than  a  rod ;  or  that 
an  acre  is  greater  or  less  than  a  degree.  Still  if  these  quan- 
tities are  expressed  by  numbers,  there  may  be  a  ratio  between 
the  numbers.  There  is  a  ratio  between  the  number  of  min- 
utes in  an  hour,  and  the  number  of  rods  in  a  mile. 

360.  Having  attended  to  the  nature  of  ratios,  we  have 
next  to  consider  in  what  manner  they  will  be  affected,  by 
varying  one  or  both  of  the  terms  between  which  the  com- 
parison is  made.  It  must  be  kept  in  mind  that,  when  a  direct 
ratio  is  expressed  by  a  fraction,  the  antecedent  of  the  couplet 
is  always  the  numerator,  and  the  consequent  the  denominator. 
It  will  be  easy,  then,  to  derive  from  the  properties  of  frac- 

*  See  Note  II. 


RATIO      AND      PROPORTION.  207 

tions,  the  changes  produced  in  ratios  by  variations  in  the 
quantities  compared.  For  the  ratio  of  the  two  quantities  is 
the  same  as  the  value  of  the  fractions,  each  being  the  quotient 
of  the  numerator  divided  by  the  denominator.  (Arts.  140, 
350.)  Now  it  has  been  shown,  (Art.  142,)  that  multiplying 
the  numerator  of  a  fraction  by  any  quantity,  is  multiplying 
the  value  by  that  quantity ;  and  that  dividing  the  numerator 
is  dividing  the  value.     Hence, 

301*  Multiplying  the  antecedent  of  a  couplet  by  any 
quantity  is  multiplying  the  ratio  by  that  quantity;  and  divid- 
ing the  antecedent  is  dividing  the  ratio. 

Thus  the  ratio  of  6  :  2,  is    3 

And  the  ratio  of  24  :  2,  is  12. 

Here  the  antecedent  and  the  ratio,  in  the  last  couplet,  are 
each  four  times  as  great  as  in  the  first. 

The  ratio  of  a  \  b,  is  r 

ii  r  .    .     na 

And  the  ratio  of  na  \  b,  is   -=-• 

b 

Cor.  With  a  given  consequent,  the  greater  the  antecedent, 
the  greater  the  ratio ;  and  on  the  other  hand,  the  greater  the 
ratio,  the  greater  the  antecedent.*     See  Art.  142,  cor. 

362.  Multiplying  the  consequent  of  a  couplet  by  any 
quantity,  is,  in  effect,  dividing  the  ratio  by  that  quantity; 
and  dividing  the  consequent  is  multiplying  the  ratio.  For 
multiplying  the  denominator  of  a  fraction,  is  dividing  the 
value  ;  and  dividing  the  denominator  is  multiplying  the  value. 
(Art.  143.) 

Thus  the  ratio  of  12  :  2>  is  6 
And  the  ratio  of    12  :  4,  is  3. 

Here  the  consequent  in  the  second  couplet,  is  twice  as 
great,  and  the  ratio  only  half  as  great,  as  in  the  first. 

The  ratio  of  a  I  b,  is  » 

o 

And  the  ratio  of  a  '.  nb,  is  — r- 

nb 

*  Euclid  8  and  10.  5.    The  first  part  of  the  propositions. 


208  RATIO      AND      PROPORTION. 

Cor.  With  a  given  antecedent,  the  greater  the  consequent, 
the  less  the  ratio  ;  and  the  greater  the  ratio,  the  less  the  con- 
sequent.#     See  Art.  143,  cor. 

363.  From  the  two  last  articles,  it  is  evident  that  multi- 
plying the  antecedent  of  a  couplet,  by  any  quantity,  will  have 
the  same  effect  on  the  ratio,  as  dividing  the  consequent  by 
that  quantity ;  and  dividing  the  antecedent,  will  have  the 
same  effect  as  multiplying  the  consequent.     See  Art.  144. 

Thus  the  ratio  of  8  :  4,  is  2 

Multiplying  the  antecedent  by  2,  the  ratio  of  16  :  4,  is  4 
Dividing  the  consequent  by  2,  the  ratio  of  8  12,  is  4. 

Cor.  Any  factor  or  divisor  may  be  transferred,  from  the 
antecedent  of  a  couplet  to  the  consequent,  or  from  the  con- 
sequent to  the  antecedent,  without  altering  the  ratio. 

It  must  be  observed  that,  when  a  factor  is  thus  transferred 
from  one  term  to  the  other,  it  becomes  a  divisor ;  and  when 
a  divisor  is  transferred,  it  becomes  a  factor. 

Thus  the  ratio  of  3X6  19=2) 

Transferring  the  factor  3,         6  :  f =2  \  the  Same  ratl°* 

_,  ma  ma  ma 

The  ratio  of  —  :  b= '-b=  -j— 

y  V  by 

Transferring  y,  ma  \  by=ma-r-by=  j—    y 

m  .  by  by      ma 

iransf  erring  m,  a  :  —  =a-f-  —  =  -r~ 

m  m       by 

364.  It  is  farther  evident,  from  Arts.  362  and  363,  that 
if  the  antecedent  and  consequent  be  both  multiplied,  or  both 
divided,  by  the  same  quantity,  the  ratio  will  not  be  altered. f 
See  Art.  145. 

Thus  the  ratio  of  8  !  4=2  ) 

Multiplying  both  terms  by  2,  16  \  8=2  >  the  same  ratio. 

Dividing  both  terms  by  2,        4  I  2=2  ) 

*  Euclid  8  and  10.  5.    The  last  part  of  the  propositions, 
f  Euclid  15.  5. 


RATIO      AND      PROPORTION.  209 

.  a 

The  ratio  of  a  I  b=r 

ma     a 

Multiplying  both  terms  by  m,       ma  \  mb—  — r  =t  y 

t^.   •  i.       ,     i  i  a     b      an      a 

Dividing  both  terms  by  n,  -  \  -  =  7-  =  7 

Cor.  1.  The  ratio  of  two  fractions  which  have  a  common 
denominator,  is  the  same  as  the  ratio  of  their  numerators. 

Thus  the  ratio  of  -  I  ->   is  the  same  as  that  of  a  I  b. 
n     n 

Cor.  2.  The  direct  ratio  of  two  fractions  which  have  a 
common  numerator,  is  the  same  as  the  reciprocal  ratio  of 
their  denominators. 

a     a     .  11 

Thus  the  ratio  of  —  I  ->   is  the  same  as  —  :  ->  or  n  :  m. 
m     n  m     n 

365.  From  the  last  article,  it  will  be  easy  to  determine 
the  ratio  of  any  two  fractions.  If  each  term  be  multiplied 
by  the  two  denominators,  the  ratio  will  be  assigned  in  inte- 
gral expressions.     Thus  multiplying  the  terms  of  the  couplet 

r  I  -.  by  bd,  we  have  -7-  :  — t-j   which  becomes  ad  :  be,  by 

cancelling  equal    quantities  from  the  numerators    and   de- 
nominators. 

300.  If  to  or  from  the  terms  of  any  couplet,  there  be 
added  or  subtracted  two  other  quantities  having  the  same 
ratio,  the  sums  or  remainders  will  also  have  the  same  ratio* 

Let  the  ratio  of  a  \b 

Be  the  same  as  that  of  c  \  d 

Then  the  ratio  of  the  sum  of  the  antecedents,  to  the  sum 
of  the  consequents,  viz.  of  a-j-c  to  b+d,  is  also  the  same. 
rT1.       .     a-\-c      c      a 

That  is  rr~^  — :}—r* 

b+d     d     b 

t^  •  •  a     c 

1.  By  supposition,  X=5 

2.  Multiplying  by  b  and  d,  ad=bc 

*  Euclid,  5  and  6.   5. 
18* 


210  RATIO      AND      PROPORTION. 

3.  Adding  cd  to  both  sides,  ad+cd=bc+cd 

bc-\-cd 

4.  Dividing  by  df  a+c= — -3 — 

a-\-c     c      a 

5.  Dividing  by  b+d,  b+d=d=b' 

The  ratio  of  the  difference  of  the  antecedents,  to  the  differ- 
ence of  the  consequents,  is  also  the  same. 

m,       .     a  —  c      c      a 

That  is 


b-d     d      b 


a     c 


1.  By  supposition,  as  before,  h~j 

2.  Multiplying  by  b  and  d,  ad=bc 
8.  Subtracting  cd  from  both  sides,      ad—cd=bc—cd 

4.  Dividing  by  d,  a  —  c= -f — 

5.  Dividing  by  b-d}  ^71=2  =  1' 

Thus  the  ratio  of  15  :  5,  is  3 

And  the  ratio  of  9  .*  3,  is  3 

Then  adding  and  subtracting  the  terms  of  the  two  couplets, 

The  ratio  of  15+9  :  5+3,  is  3 

And  the  ratio  of  15— 9  :  5— 3,  is  3 

Here  the   terms  of  only  two  couplets  have  been   addea 

together.     But  the  proof  may  be  extended  to  any  number  of 

couplets  where  the  ratios  are  equal.     For,  by  the  addition  of 

the  two  first,  a  new  couplet  is  formed,  to  which,  upon  the 

same  principle,  a  third  may  be  added,  a  fourth,  &c.     Hence, 

367.  If,  in  several  couplets,  the  ratios  are  equal,  the  sum 
of  all  the  antecedents  has  the  same  ratio  to  the  sum  of  all  the 
consequents,  which  any  one  of  the  antecedents  has  to  its  con- 
sequent.* 

fl2  :  6=2 

10 :  5=2 
8 :  4=2 
6 :  3=2 

Therefore  the  ratio  of  (12  +  10+8+6)  ;  (6+5+4+3)=2. 
*  Euclid  1  and  12.  5. 


Thus  the  ratio 


eatio    and    proportion.  211 

Ratios  of  Inequality. 

368.  A  ratio  of  greater  inequality  is  diminished,  by 
adding  the  same  quantity  to  both  the  terms. 

Let  the  given  ratio  be  that  of  a+b  l  a  or  

D  a 

Adding  x  to  both  terms,  it  becomes  a+b+x :  a+x  or 

a+x 

Reducing  them  to  a  common  denominator, 

rp,     a    A  i  a2+ab+ax+bx 

1  he  first  becomes  ; r 

a(a+x) 

And  the  latter,  ; — . 

a(a+x) 

As  the  latter  numerator  is  manifestly  less  than  the  other, 
the  ratio  must  be  less.     (Art.  362,  cor.) 

But  a  ratio  of  lesser  inequality  is  increased,  by  adding  the 
same  quantity  to  both  terms. 

Let  the  given  ratio  be  that  of  a—b'.a,   or   

a 

a-b+x 


Adding  x  to  both  terms,  it  becomes  a-b+x  :  a+x  or 

a+x 

Reducing  them  to  a  common  denominator, 

rp,     c       ,  a2—ab+ax—bx 

lhe  first  becomes  - r 

a(a+x) 

.    .     .  a2  —ab  +  ax 

And  the  latter,  5 — 

a(a+x) 

As  the  latter  numerator  is  greater  than  the  other,  the  ratio 
is  greater. 

On  the  other  hand,  a  ratio  of  greater  inequality  is  increased, 
and  one  of  lesser  inequality  is  diminished,  by  subtracting 
from  both  the  terms  any  quantity  less  than  either  of  them. 

309.  If  the  same  quantity  be  added  to  the  two  terms  of 
an  inequality,  the  ratio  will  not  be  changed  from  a  greater 
to  a  lesser  inequality,  or  the  contrary;  but  will  subsist  in  the 
same  sense,  after  the  addition. 

If  in  the  ratio  of  c  :  d,  the  antecedent  c  is  greater  than 
the  consequent  d,  then  it  is  evident  that   c+x   is  greater 


212  RATIO      AND      PROPORTION. 

than  d+x,  (Art.  58,  ax.  7.)  Though  the  ratio,  according  to 
the  preceding  article,  is  diminished  by  the  addition  of  x,  yet 
it  is  not  changed  to  a  contrary  inequality. 

If  in  the  ratio  of  m  \  n,  m  is  less  than  n,  it  is  evident  that 
m +x  is  less  than  n+x.  Though  the  ratio  is  increased,  by  the 
addition  of  x,  yet  it  is  not  changed  to  a  contrary  inequality. 

If  the  corresponding  terms  of  two  ratios,  subsisting  in  the 
same  sense,  be  added  together,  the  ratio  will  not  be  changed 
to  a  contrary  inequality.  If  the  antecedent  a  is  greater  than 
the  consequent  b,  and  the  antecedent  c  is  greater  than  the 
consequent  d,  it  is  manifest  that  a+c  is  greater  than  b+d. 
And  it  e  is  less  than  f,  and  g  less  than  h,  e+g  is  evidently 
less  than  f+h. 

370.  A  ratio  of  greater  inequality,  compounded  with 
another  ratio,  increases  it. 

Let  the  ratio  of  greater  inequality  be  that  of  1+n  :  1 

And  any  given  ratio,  that  of  a  '.  b 

The  ratio  compounded  of  these,  (Art.  357,)  is       a+na  :  b 
Which  is  greater  than  that  of  a  :  b  (Art.  362,  cor.) 
But  a  ratio  of  lesser  inequality,  compounded  with  another 
ratio,  diminishes  it. 

Let  the  ratio  of  lesser  inequality  be  that  of 
And  any  given  ratio  that  of 

The  ratio  compounded  of  these  is 
Which  is  less  than  that  of  a  '.  b. 
Thus  the  ratio  of 
That  of 

The  ratio  compounded  of  these  is  60  !  6=10. 
A  ratio  of  greater  inequality  compounded  with  one  of 
lesser  inequality  may  increase  the  latter  so  much,  as  to  con- 
vert it  into  one  of  greater  inequality.^ 

The  ratio  of  2  :  4,  is  \ 

That  of  8  :  2,  is  4 

The  compound  ratio,  is  16  :  8=2 

On  the  other  hand,  a  ratio  of  lesser  inequality  compounded 
with  one  of  greater  inequality,  may  diminish  the  latter  so 
much,  as  to  convert  it  into  one  of  less  inequality. 


hat 

of            1—n  : 

1 

a  : 

:  b 

a—na  ! 

:b 

6 

!  3,  is  2  ) 

10 

!  2,  is  5  ) 

RATIO      AND      PROPORTION.  213 

Examples. 

1.  Which  is  the  greatest,  the  ratio  of  11  I  9,  or  that  of 
44  :  35  ? 

2.  Which  is  the  greatest,  the  ratio  of  a+3  :  \a,  or  that 
of  2a+l  \\al 

3.  If  the  antecedent  of  a  couplet  be  G5y  and  the  ratio  13, 
what  is  the  consequent  ? 

4.  If  the  consequent  of  a  couplet  be  7,  and  the  ratio  18, 
what  is  the  antecedent  ? 

5.  What  is  the  ratio  compounded  of  the  ratios  of  3  !  7, 
and  2a  :  56,  and  7x+l  [  Sy-2? 

6.  What  is  the  ratio  compounded  of  x+y  '.  b,  and 
x—y  !  a+b,  and  a-\-b  \  h?  Ans.  x2—y2  '.  bh. 

7.  If  the  ratios  of  5^+7  :  2a; -3,  and  x+2  \  \x+ 3  be 
compounded,  will  they  produce  a  ratio  of  greater  inequality,  or 
of  lesser  inequality  ?        Ans.  A  ratio  of  greater  inequality. 

8.  What  is  the  ratio  compounded  of  x+y  '.  a,  and  x-y  I  b, 

and  b  '. ?  Ans.   A  ratio  of  equality. 

9.  What  is  the  ratio  compounded  of  7  I  5,  and  the  dupli- 
cate ratio  of  4  :  9,  and  the  triplicate  ratio  of  3  :  2  ? 

Ans.  14  :  15. 

10.  What  is  the  ratio  compounded  of  3  :  7,  and  the  tripli- 
cate ratio  of  x  I  y,  and  the  subduplicate  ratio  of  49  !  9? 

Ans.  x3  I  y3. 

Proportion. 

•171.  An  accurate  and  familiar  acquaintance  with  the 
doctrine  of  ratios,  is  necessary  to  a  ready  understanding  of 
the  principles  of  proportion,  one  of  the  most  important  of  all 
the  branches  of  the  mathematics.  In  considering  ratios,  we 
compare  two  quantities,  for  the  purpose  of  finding  either  their 
difference,  or  the  quotient  of  the  one  divided  by  the  other. 
But  in  proportion,  the  comparison  is  between  two  ratios. 
And  this  comparison  is  limited  to  such  ratios  as  are  equal 
We  do  not  inquire  how  much  one  ratio  is  greater  or  less  than 
another,  but  whether  they  are  the  same.  Thus  the  numbers 
12,  6,  8,  4,  are  said  to  be  proportional,  because  the  ratio  of 
12  :  6  is  the  same  as  that  of  8  :  4. 


214  RATIO      AND      PROPORTION. 

372.  Proportion,  then,  is  an  equality  of  ratios.  It  is 
either  arithmetical  or  geometrical.  Arithmetical  proportion 
is  an  equality  of  arithmetical  ratios,  and  geometrical  propor- 
tion is  an  equality  of  geometrical  ratios.*  Thus  the  numbers 
6,  4,  10,  8,  are  in  arithmetical  proportion,  because  the  differ- 
ence between  6  and  4  is  the  same  as  the  difference  between 
10  and  8.  And  the  numbers  6,  2,  12,  4,  are  in  geometrical 
proportion,  because  the  quotient  of  6  divided  by  2,  is  the 
same  as  the  quotient  of  12  divided  by  4. 

373.  Care  must  be  taken  not  to  confound  proportion  with 
ratio.  This  caution  is  the  more  necessary,  as  in  common 
discourse,  the  two  terms  are  used  indiscriminately,  or  rather, 
proportion  is  used  for  both.  The  expenses  of  one  man  are 
said  to  bear  a  greater  proportion  to  his  income,  than  those  of 
another.     But  according  to  the  definition  which  has  just  been 

fiven,  one  proportion  is  neither  greater  nor  less  than  another, 
'or  equality  does  not  admit  of  degrees.  One  ratio  may  be 
greater  or  less  than  another.  The  ratio  of  12  :  2  is  greater 
than  that  of  6  :  2,  and  less  than  that  of  20  !  2.  But  these 
differences  are  not  applicable  to  proportion,  when  the  term  is 
used  in  its  technical  sense.  The  loose  signification  which  is 
so  frequently  attached  to  this  word,  may  be  proper  enough  in 
familiar  language:  for  it  is  sanctioned  by  a  general  usage. 
But  for  scientific  purposes,  the  distinction  between  proportion 
and  ratio  should  be  clearly  drawn,  and  cautiously  observed. 

374.  The  equality  between  two  ratios,  as  has  been  stated, 
is  called  proportion.  The  word  is  sometimes  applied  also 
to  the  series  of  terms  among  which  this  equality  of  ratios 
exists.  Thus  the  two  couplets  15  :  5  and  6  .'  2  are,  when 
taken  together,  called  a  proportion. 

375.  Proportion  may  be  expressed,  either  by  the  common 
sign  of  equality,  or  by  four  points  between  the  two  couplets. 

8- -6=4- -2,    or    8-  -6:  !4-  -2  )  are  arithmetical 


Thus 

•a,    or    a  —  b.mC'-d)    proportions. 


a- 

•b-- 

=s  C 

12  : 

:  6= 

=  8 

a  : 

:b= 

=d 

And       I  12  '  6=8  '  4'  °r  12  '  6!  !8  '  4  \  are  geometrical 
I    a  :  b=d  :  h}  or     a\b\\d\h)    proportions. 

The  latter  is  read,  '  the  ratio  of  a  to  b  equals  the  ratio  of 
d  to  h\    or  more  concisely,  'a  is  to  b,  as  d  to  h.f 

*  See  Note  L 


RATIO      AND      PROPORTION.  215 

376.  The  first  and  last  terms  are  called  the  extremes,  and 
the  other  two  the  means.  Homologous  terms  are  either  the 
two  antecedents  or  the  two  consequents.  Analogous  terms 
are  the  antecedent  and  consequent  of  the  same  couplet. 

377.  As  the  ratios  are  equal,  it  is  manifestly  immaterial 
which  of  the  two  couplets  is  placed  first. 

a     c  c      a 

If  a\b\\c  I  d,  then  c  I  d\  \a  \  b.     For  if  T= -.  then  -7=y 

b     a  a      b 

378.  The  number  of  terms  must  be,  at  least,  four.  For 
the  equality  is  between  the  ratios  of  two  couplets;  and  each 
couplet  must  have  an  antecedent  and  a  consequent.  There 
may  be  a  proportion,  however,  among  three  quantities.  For 
one  of  the  quantities  may  be  repeated,  so  as  to  form  two 
terms.  In  this  case  the  quantity  repeated  is  called  the  middle 
term,  or  a  mean  proportional  between  the  two  other  quanti- 
ties, especially  if  the  proportion  is  geometrical. 

Thus  the  numbers  8,  4,  2,  are  proportional.  That  is, 
8  I  4;  ;4  I  2.  Here  4  is  both  the  consequent  in  the  first 
couplet,  and  the  antecedent  in  the  last.  It  is  therefore  a 
mean  proportional  between  8  and  2. 

The  last  term  is  called  a  third  proportional  to  the  two 
other  quantities.     Thus  2  is  a  third  proportional  to  8  and  4. 

379.  Inverse  or  reciprocal  proportion  is  an  equality  be- 
tween a  direct  ratio,  and  a  reciprocal  ratio. 

Thus  4  :  2!  *.^  !  £ ;  that  is,  4  is  to  2,  reciprocally,  as  3  to 
6.  Sometimes  also,  the  order  of  the  terms  in  one  of  the 
couplets,  is  inverted,  without  writing  them  in  the  form  of  a 
fraction.     (Art.  355.) 

Thus  4  \  2!  ;3  \  6  inversely.  In  this  case,  the  first  term  is 
to  the  second,  as  the  fourth  to  the  third;  that  is,  the  first  divi- 
ded by  the  second,  is  equal  to  the  fourth  divided  by  the  third. 

380.  When  there  is  a  series  of  quantities,  such  that  the 
ratios  of  the  first  to  the  second,  of  the  second  to  the  third, 
of  the  third  to  the  fourth,  &c.  are  all  equal;  the  quantities 
are  said  to  be  in  continued  proportion.  The  consequent  of 
each  preceding  ratio  is,  then,  the  antecedent  of  the  following 
one. — Continued  proportion  is  also  called  progression,  as  will 
be  seen  in  a  following  section. 

Thus  the  numbers  10,  8,  6,  4,  2,  are  in  continued  arith- 
metical proportion.     For  10  —  8=8—6=6—4=4—2. 


216  RATIO      AND      PROPORTION. 

The  numbers  64,  32,  16,  8,  4,  are  in  continued  geometrical 
proportion.     For  64  :  32!  |32  :  16:  !16  :  8!  18  :  4. 

If  a,  b,  c,  d,  h,  &c.  are  in  continued  geometrical  propor- 
tion ;  then  a  \  b\\b  \  c\\c  \  d\  \d  I  h,  &c. 

One  case  of  continued  proportion  is  that  of  three  propor- 
tional quantities.     (Art.  378.) 

381.  As  an  arithmetical  proportion  is,  generally,  nothing 
more  than  a  very  simple  equation,  it  is  scarcely  necessary  to 
give  the  subject  a  separate  consideration. 

The  proportion  a--b\\C"d 

Is  the  same  as  the  equation         a—b=c—d 

It  will  be  proper,  however,  to  observe  that,  {{four  quanti- 
ties are  in  arithmetical  proportion,  the  sum  of  the  extremes  is 
equal  to  the  sum  of  the  means. 

Thus  if  a-  -bWh  —  m,  then  a+m=b+h 

For  by  supposition,  a—b=h—m 

And  transposing  —b  and  —  m,  a+m=b+h 

So  in  the  proportion,  12- -10!  Ill  •  -9,  we  have  12+9=10  +  11. 

Again  if  three  quantities  are  in  arithmetical  proportion, 
the  sum  of  the  extremes  is  equal  to  double  the  mean. 

If  a  —  b\  \b-  -c,  then  a—b=b—c 

And  transposing  —b  and  — c,  a+c=2b. 

Geometrical  Proportion. 

38£.  But  if  four  quantities  are  in  geometrical  proportion, 
the  product  of  the  extremes  is  equal  to  the  product  of  the  means. 

If  a I  b*. "■•<?  J  d,  ad=^bc 

a      c 

b  =  d 
ahd     cbd 

~b~  =  ~d' 
Reducing  the  fractions,  ad=bc 

Thus  12  :  8!  !15  I  10,  therefore  12X10=8X15. 
Cor.  Any  factor  may  be  transferred  from  one  mean  to  the 
other,  or  from  one  extreme  to  the  other,  without  affecting 
the  proportion.  If  a  \  mb\  \x  \  y,  then  a  \  b\  \mx  \  y.  For 
the  product  of  the  means  is,  in  both  cases,  the  same.  And 
if  na  \  b\  \x  \  y}  then  a  \  b\  \x  '  ny. 


For  by  supposition,  (Arts.  350,  372,) 
Multiplying  by  bd9  (Ax.  3,) 


m 

h 

—  =z 

:  — 

n 

y 

my 

nh 

ny 

~  ny 

RATIO      AND      PROPORTION.  217 

383.  On  the  other  hand,  if  the  product  of  two  quantities 
is  equal  to  the  product  of  two  others,  the  four  quantities  will 
form  a  proportion,  when  they  are  so  arranged,  that  those  on 
one  side  of  the  equation  shall  constitute  the  means,  and 
those  on  the  other  side,  the  extremes. 

If  my=nh,  then  m\n\\h\y,  that  is, 

For  by  dividing  my=nh  by  ny,  we  have 

And  reducing  the  fractions,  —  =  -  • 

n     y 

Cor.  The  same  must  Ke^true  of  any  factors  which  form 
the  two  sides  of  an  equation. 

If  (a-\-b) Xc=  (d—m) Xy,  then  a+b  I  d—m\  \y  \  c. 

384.  If  three  quantities  are  proportional,  the  product  of 
the  extremes  is  equal  to  the  square  of  the  mean.  For  this 
mean  proportional  is,  at  the  same  time,  the  consequent  of  the 
first  couplet,  and  the  antecedent  of  the  last.  (Art.  378.)  It  is 
therefore  to  be  multiplied  into  itself,  that  is,  it  is  to  be  squared. 

If  a\b\\b  :  c,  then  multiply  extremes  and  means,  ac=b2. 
Hence,  a  mean  proportional  between  two  quantities  may 
be  found,  by  extracting  the  square  root  of  their  product. 
If  a  :  x\  \x  :  c,  then  x2—ac,  and  x=y/ac.     (Art.  314.) 

385.  It  follows,  from  Art.  383,  that  in  a  proportion,  either 
extreme  is  equal  to  the  product  of  the  means,  divided  by  the 
other  extreme ;  and  either  of  the  means  is  equal  to  the  pro- 
duct of  the  extremes,  divided  by  the  other  mean. 

1.  If  a  :  b\  \c  :  d,  then  ad— be 

be 

2.  Dividing  by  d,  a~~l 

3.  Dividing  the  first  by  c,  &=  — 

4.  Dividing  it  by  b,  c=  -j? 

be 

5.  Dividing  it  by  a,  d=—  \     that    is,    the 

fourth  term  is  equal  to  the  product  of  the  second  and  third 
divided  by  the  first. 

19 


218  RATIO      AND      PROPORTION. 

On  this  principle  is  founded  the  rule  of  simple  proportion 
in  arithmetic,  commonly  called  the  Rule  of  Three.  Three 
numbers  are  given  to  find  a  fourth,  which  is  obtained  by 
multiplying  together  the  second  and  third,  and  dividing  by 
the  first. 

380.  The  propositions  respecting  the  products  of  the 
means,  and  of  the  extremes,  furnish  a  very  simple  and  con- 
venient criterion  for  determining  whether  any  four  quantities 
are  proportional.  We  -have  only  to  multiply  the  means 
together,  and  also  the  extremes.  If  the  products  are  equal, 
the  quantities  are  proportional.  If  the  products  are  not 
equal,  the  quantities  are  not  projjortional. 

387.  In  mathematical  investigations,  when  the  relations 
of  several  quantities  are  given,  they  are  frequently  stated  in 
the  form  of  a  proportion.  But  it  is  commonly  necessary  that 
this  first  proportion  should  pass  through  a  number  of  trans- 
formations before  it  brings  out  distinctly  the  unknown  quan- 
tity, or  the  proposition  which  we  wish  to  demonstrate.  It 
may  undergo  any  change  which  will  not  affect  the  equality 
of  the  ratios ;  or  which  will  leave  the  product  of  the  means 
equal  to  the  product  of  the  extremes. 

It  is  evident,  in  the  first  place,  that  any  alteration  in  the 
arrangement,  which  will  not  affect  the  equality  of  these  two 
products,  will  not  destroy  the  proportion.  Thus,  \f  a:b::  c:d, 
the  order  of  these  four  quantities  may  be  varied,  in  any  way 
which  will  leave  ad— be.     Hence, 

388.  If  four  quantities  are  proportional,  the  order  of  the 
means,  or  of  the  extremes,  or  of  the  terms  of  both  couplets, 
may  be  inverted  without  destroying  the  proportion. 

If  a:b  : :  c:  <i >  , 

And         12:8::6:4S         ' 
1.  Inverting  the  means* 

a\  c\\b  \  d  )  (  The  first  is  to  the  third, 

12  I  6!  18  :  4  )  (As  the  second  to  the  fourth. 

In  other  words,  the  ratio  of  the  antecedents  is  equal  to  the 
ratio  of  the  consequents. 

This  inversion  of  the  means  is  frequently  referred  to  by 
geometers,  under  the  name  of  Alternation.^ 

*  See  Note  K.  •)•  Euclid,  16.  5. 


RATIO      AND      PROPORTION.  219 

2.  Inverting  the  extremes, 

d  \b\\c  \  a    j    ,        .e  j  The  fourth  is  to  the  second, 


4  :  8!  !6  :  12  )  '  (  As  the  third  to  the  first 

3.  Inverting  the  terms  of  each  couplet , 

b  *    a]  !^  .'  c  )    ,       .     (The  second  is  to  the  ^rs/, 
8  I  12!  !4  :  6  )  t  at  1S'  (As  the  /oar**  to  the  third. 

This  is  technically  called  Inversion. 

Each  of  these  may  also  be  varied,  by  changing  the  order 
of  the  two  couplets.    ^Art.  378.) 

Cor.  The  order  of  the  whole  proportion  may  be  inverted. 
If  a  \  b\  \c  l  d,  then  d  '.  c\  \b  \  a. 

In  each  of  these  cases,  it  will  be  at  once  seen  that,  by 
taking  the  products  of  the  means,  and  of  the  extremes,  we 
have  ad=bc,  and  12X4=8X6. 

If  the  terms  of  only  one  of  the  couplets  are  inverted,  the 
proportion  becomes  reciprocal.     (Art.  379.) 

If  a  l  b\  \c  I  d,  then  a  is  to  b,  reciprocally,  as  d  to  c. 

380*  A  difference  of  arrangement  is  not  the  only  altera- 
tion which  we  have  occasion  to  produce,  in  the  terms  of  a 
proportion.  It  is  frequently  necessary  to  multiply,  divide,  in- 
volve, &c.  In  all  cases,  the  art  of  conducting  the  investiga- 
tion consists  in  so  ordering  the  several  changes,  as  to  maintain 
a  constant  equality,  between  the  ratio  of  the  two  first  terms, 
and  that  of  the  two  last.  As  in  resolving  an  equation,  we 
must  see  that  the  sides  remain  equal ;  so  in  varying  a  pro- 
portion,  the  equality  of  the  ratios  must  be  preserved.  And 
this  is  effected  either  by  keeping  the  ratios  the  same,  while 
the  terms  are  altered  ;  or  by  increasing  or  diminishing  one  of 
the  ratios  as  much  as  the  other.  Most  of  the  succeeding  proofs 
are  intended  to  bring  this  principle  distinctly  into  view,  and 
to  make  it  familiar.  Some  of  the  propositions  might  be  de- 
monstrated, in  a  more  simple  manner,  perhaps,  by  multiplying 
the  extremes  and  means.  But  this  would  not  give  so  clear  a 
view  of  the  nature  of  the  several  changes  in  the  proportions. 

It  has  been  shown  that,  if  both  the  terms  of  a  couplet  be 
multiplied  or  divided  by  the  same  quantity,  the  ratio  will  re- 
main the  same ;  (Art.  364,)  that  multiplying  the  antecedent 
is,  in  effect,  multiplying  the  ratio,  and  dividing  the  antece- 
dent, is  dividing  the  ratio;  (Art.  361,)  and  farther,  that  mub 
tiplying  the  consequent,  is,  in  effect,  dividing  the  ratio,  and 


220 


RATIO      AND      PROPORTION. 


dividing  the  consequent  is  multiplying  the  ratio.  (Art.  362.) 
As  the  ratios  in  a  proportion  are  equal,  if  they  are  both 
multiplied,  or  both  divided,  by  the  same  quantity,  they  will 
still  be  equal.  (Ax.  3.)  One  will  be  increased  or  diminished 
as  much  as  the  other.     Hence, 

390.  If  four  quantities  are  proportional,  two  analogous 
or  two  homologous  terms  may  be  multiplied  or  divided  by  the 
same  quantity  without  destroying  the  proportion. 

If  analogous  terms  be  multiplied  or  divided,  the  ratios  will 
not  be  altered.  (Art.  364.)  If  homologous  terms  be  multi- 
plied or  divided,  both  ratios  will  be  equally  increased  or 
diminished.     (Arts.  361,  2.) 

If  a:b::c:d,  then, 

1.  Multiplying  the  two  first  terms,  ma\mb\\c\  d 

2.  Multiplying  the  two  last  terms,  a  \  b\  \mc  \  md 

3.  Multiplying  the  two  antecedents,*  ma  \b\\mc\  d 

4.  Multiplying  the  two  consequents,  a  \  mb\  \c  '.  md 

a 


5.  Dividing  the  two  first  terms, 

6.  Dividing  the  two  last  terms, 

7.  Dividing  the  two  antecedents, 

8.  Dividing  the  two  consequents, 


m 


b 
m 


m 
c 


-:&::- 


d 

A 

m 

Id 

d 
m 


Cor.  1.  All  the  terms  may  be  multiplied  or  divided  by  the 

same  quantity. j* 

_    a      b      c      d 
ma  :  mb\  \mc  \  md,  —  \—\\—  \  —• 

m     m     m     m 

Cor.  2.  In  any  of  the  cases  in  this  article,  multiplication 
of  the  consequent  may  be  substituted  for  division  of  the 
antecedent  in  the  same  couplet,  and  division  of  the  conse- 
quent, for  multiplication  of  the  antecedent.    (Art.  363,  cor.) 


* 

3 

r   h 

- 

d 

Thus 

ma  :  b : 

:  mc :  d 

a  :  — : 
m 

:  mc  :  d 

ma  :  b 

:  c  :  -— 
m 

for  ' 

a     i 

c 

"   ST  ' 

CD 

c 

►  or  < 

a 

-  :  b  : 

:  -  :  d 

^ 

a  :  mb  : 

:-:d 

-:b: 

c  :  md 

m 

m 

g 

m 

m 

*  Euclid,  8.  6. 


f  Euclid,  4.  5. 


RATIO      AND      PROPORTION.  221 

39 1 .  It  is  often  necessary  not  only  to  alter  the  terms  of 
a  proportion,  and  to  vary  the  arrangement,  but  to  compare 
one  proportion  with  another.  From  this  comparison  will 
frequently  arise  a  new  proportion,  which  may  be  requisite  in 
solving  a  problem,  or  in  carrying  forward  a  demonstration. 
One  of  the  most  important  cases  is  that  in  which  two  of  the 
terms  in  one  of  the  proportions  compared,  are  the  same  with 
two  in  the  other.  The  similar  terms  may  be  made  to  dis- 
appear, and  a  new  proportion  may  be  formed  of  the  four 
remaining  terms.     For, 

392.  If  two  ratios  are  respectively  equal  to  a  third,  they 
are  equal  to  each  other  * 

This  is  nothing  more  than  the  11th  axiom  applied  to  ratios. 

1.  If  a:  b:  :m  :  n  )    ,  7  7  ,     *    ,  *  v 

A     ,         y  {  then  a:b:  :c:d,  or  a:  c::b:d.  (Art.  388.) 

And   c :  d:  :m:  n  )  v  ' 


2.  If       a  :  b\  :m  :  n) 

.     ,  7  f  then  a:  b:\c\d,  or  a:  c:\b  :  d. 

And  m:n:.:c  :  d) 

Cor.  If         a:b::m:n)    .  ,.  ,, 

.     .  .  7  f  then  a  :  bj>c  :  d.r 

And    m  :  ?f>c :  :  d  ) 

For  if  the  ratio  of  m  ;  n  is  greater  than  that  of  c  ;  d,  it 
is  manifest  that  the  ratio  of  a  I  6,  which  is  equal  to  that  of 
m  '  n,  is  also  greater  than  that  of  c  *  d. 

393.  In  these  instances,  the  terms  which  are  alike  in  the 
two  proportions  are  the  two  first  arid  the  two  last.  But  this 
arrangement  is  not  essential.  The  order  of  the  terms  may 
De  changed  in  various  ways,  without  affecting  the  equality 
of  the  ratios. 

1.  The  similar  terms  may  be  the  two  antecedents,  or  the 
two  consequents,  in  each  proportion.     Thus, 

If       m:  a:\n\b\     .         (By  alternation,    m:n:  :a:b 
And  m:  c::n:  d)  \  And  m:n::c  :  d 

Therefore  a:b:  :c  :  dt  or  a:  c:  :b:  d,  by  the  last  article. 

2.  The  antecedents  in  one  of  the  proportions,  may  be  the 
same  as  the  consequents  in  the  other. 

If      m:  a:  :n:b\  (By  inver.  and  altern.    a:b:  :m:  n 

And  cmwd'.n)  (By  alternation,  c:  d::m:n 

Therefore  a  •  b,  &c.  as  before. 

t  Euclid,  11.  §.  f  Euclid,  13.  6. 

19* 


222  RATIO      AND      PROPORTION. 

3.  Two  homologous  terms,  in  one  of  the  proportions,  may 
be  the  same,  as  two  analogous  terms  in  the  other. 

If      a  :  m\  \b  :  n  )  (By  alternation,  a  :  b\  \m  \  n 

And  c  :  d\  \m  \  n  )       en   (  And  c  :  d\  \m\  n 

Therefore,  a  \  b,  &c. 

All  these  are  instances  of  an  equality,  between  the  ratios 
in  one  proportion,  and  those  in  another.  In  geometry,  the 
proposition  to  which  they  belong  is  usually  cited  by  the 
words  "  ex  aequo,"  or  "  ex  aequali."*  The  second  case  in 
this  article  is  that  which  in  its  form,  most  obviously  answers 
to  the  explanation  in  Euclid.  But  they  are  all  upon  the 
same  principle,  and  are  frequently  referred  to,  without  dis- 
crimination. 

394.  Any  number  of  proportions  may  be  compared,  in 
the  same  manner,  if  the  two  first  or  the  two  last  terms  in 
each  preceding  proportion,  are  the  same  with  the  two  first 
or  the  two  last  in  the  following  one.* 

Thus  if    a\b\\c  '.  d  \ 

And  c  :  d\\h  \  I  (   ,  -  •      . 

a    j  7.7..      .  ™  }  then  a  .  b.  .x  .  y. 

And  h  .  1.  ,m  .  n(  y 

And         m  \  n\  \x  \  y) 

That  is,  the  two  first  terms  of  the  first  proportion  have  the 
same  ratio,  as  the  two  last  terms  of  the  last  proportion.  For 
it  is  manifest  that  the  ratio  of  all  the  couplets  is  the  same. 

And  if  the  terms  do  not  stand  in  the  same  order  as  here, 
yet  if  they  can  be  reduced  to  this  form,  the  same  principle  is 
applicable. 

Thus  if  a  '.  c\  \b  \  d 

And        c  :  h\  \d  \  I 

And        h  \  m\\l  \  n 

And       m  \  x\\n  \  y j 

Therefore  a  \b\\x  \  y,  as  before. 

In  all  the  examples  in  this,  and  the  preceding  articles,  the 
two  terms  in  one  proportion  which  have  equals  in  another, 
are  neither  the  two  means,  nor  the  two  extremes,  but  one  of 
the  means,  and  one  of  the  extremes ;  and  the  resulting  pro- 
portion is  uniformly  direct. 

*  Euclid,  22.  5. 


then  bv  alternation 


fa 

:b: 

\c  : 

d 

c 

:  d: 

:*: 

I 

h 

:  i: 

\m 

\  n 

\m 

'.  n\ 

\x 

•  9 

n 

m 

\  a\ 

\o : 

n 

And 

m 

\  c\ 

\d\ 

n 

Or  if 

a  \ 

m\ 

\n 

:  b 

And 

m  : 

\  c  \ 

\d\ 

;  n 

RATIO      AND      PROPORTION.  223 

39*>.  But  if  the  two  means,  or  the  two  extremes,  in  one 
proportion,  be  the  same  with  the  means,  or  the  extremes, 
in  another,  the  four  remaining  terms  will  be  reciprocally 
proportional. 

if     a :  m:  :* :  6 )  ■  1    1 

a    j       .     .  .     .  j  X  then  a  \  c\  ;r  \  ->  or  a  \  c\  \d  I  b. 
And    c  \  m.  ,n  .  a  )  b     a 

or  a  —  n  /Art  382  \  Therefore  ab=cd,  and  a\  c\\d\b. 
And  cd=?nn  )  v  ' 

In  this  example,  the  two  means  in  one  proportion,  are  like 
those  in  the  other.  But  the  principle  will  be  the  same,  if 
the  extremes  are  alike,  or  if  the  extremes  in  one  proportion 
are  like  the  means  in  the  other. 

then  a  \  c\\d  \b. 

then  a  \  c\  \ d  \  b. 

The  proposition  in  geometry  which  applies  to  this  case,  is 
usually  cited  by  the  words  "  ex  aequo  perturbate."* 

390.  Another  way  in  which  the  terms  of  a  proportion 
may  be  varied,  is  by  addition  or  subtraction. 

If  to  or  from  two  analogous  or  two  homologous  terms  of 
a  proportion,  two  other  quantities  having  the  same  ratio  be 
added  or  subtracted,  the  proportion  will  be  preserved.^ 

For  a  ratio  is  not  altered,  by  adding  to  it,  or  subtracting 
from  it,  the  terms  of  another  equal  ratio.     (Art.  366.) 

If         a  :  b  : :  c  :  d  ) 
And    a:b::m:n  5 

Then  by  adding  to,  or  subtracting  from  a  and  b,  the 
terms  of  the  equal  ratio  m  \  n,  we  have, 

a+m  :  b+n\  \c  \  d,         and  a—m  \  b—n\  \c  \  d. 
And  by  adding  and  subtracting  m  and  n,  to  and  from  c 
and  d  we  have, 

a.  I  b\  \c+m  l  d+n,         and  a  \  b\  \c  —  m  \  d—n. 
Here  the  addition  and  subtraction  are  to  and  from  analo- 
gous terms.     But  by  alternation,  (Art.  388,)  these  terms  will 
become  homologous,  and  we  shall  have, 

a+m  :  c\  \b-\-n  \  d,         and  a—m  \  c\  \b—n  \  d. 

*  Euclid,  23.  5.  f  Euclid,  2.  5. 


224  RATIO      AND      PROPORTION 

Cor.  1.  This  addition  may,  evidently,  be  extended  to  any 
number  of  equal  ratios.* 

r  c  \  d 

Thus  if  a  :  b: :  <  * ' ; l 

jm  .  n 

^x'.y 
Then  a  \  b\  'mc+h+m+x  I  d+l+n+y. 
Cor.  2.  If        a\b\\c\  d 


And  m  \  b\ 


c  '  d  ) 

.  [  then  a+m  :  b\  \c+n  \  d.i 
n  :  d  ) 


For  by  alternation  a\  c\\b  \  d)  there-  C      a+m  \  c+n\  \b  yd 
And  m\n\\b\d)    f°re  (or  a+m  \  b\  \c+n  :  d. 

397.  From  the  last  article  it  is  evident  that  if,  in  any 
proportion,  the  terms  be  added  to,  or  subtracted  from  each 
other,  that  is, 

If  two  analogous  or  homologous  terms  be  added  to,  or  sub- 
tracted from  the  two  others,  the  proportion  will  be  preserved. 

Thus,  if  a:b::c:  d,  and   12  :  4  : :  6  :  2,  then, 

1 .  Adding  the  two  last  terms,  to  the  two  first. 

a+c:b+d::a:b         12+6:  4+2-12:4 

and  a+c  :  b+d\  \c  \  d  12+6:    4+2:  I  6  :  2 

or  a+c  :  a:  \b+d  :  b  12+6  :  12:  :4+  2:4 

and  a+c  *  c\  \b+d  \  d  12+6  :    6\  \4+  2  :  2. 

2.  Adding  the  two  antecedents,  to  the  two  consequents. 

a+b\b\\c+d\d  12+4:    4]  :6+2  :  2 

«+& :  a:  :c+d :  c,  &c.     12+4  :  12:  :6+2  :  6,  &o, 

This  is  called  Composition^ 

3.  Subtracting  the  two  jftrsJ  terms,  from  the  two  last. 

c—a  :  #:  \d—b :  & 

c— <z  :  c\  \d—b  :  c?,  &c. 

4.  Subtracting  the  two  /#$£  terms  from  the  two  ^rs£. 

a— c :  6— ^:  \a  :  &§ 

a— c  :  b-  d\  \c  ;  c?,  &c. 

*  Euclid,  2.  5.     Cor.  +  Euclid,  24.  5. 

\  Euclid,  18.  5.  g  Euclid,  19.  5. 


RATIO      AND      PROPORTION.  225 

5.  Subtracting  the  consequents  from  the  antecedents. 

a—b  I  b\  \c—d  I  d 

a  \  a—b'.  \c  \  c—d,  &c. 

The  alteration  expressed  by  the  last  of  these  forms  is  called 
Conversion. 

6.  Subtracting  the  antecedents  from  the  consequents. 

b—a  ;  a\  \d—c  \  c 

b  \  b—a\  \d  \  d—c,  &c. 

7.  Adding  and  subtracting, 

a+b  :  a—b\  \c~\-d  \  c—d. 

That  is,  the  sum  of  the  two  first  terms,  is  to  their  differ- 
ence, as  the  sum  of  the  two  last,  to  their  difference. 

Cor.  If  any  compound  quantities,  arranged  as  in  the  pre- 
ceding examples,  are  proportional,  the  simple  quantities  of 
which  they  are  compounded  are  proportional  also. 

Thus,  if  a+b  :  b\  \c+d  \  d}  then  a\b\\c  \  d. 
This  is  called  Division* 

308.  If  the  corresponding  terms  of  two  or  more  ranks  of 
proportional  quantities  be  multiplied  together,  the  product 
will  be  proportional. 

This  is  compounding  ratios,  (Art.  356,)  or  compounding 
proportions.  It  should  be  distinguished  from  what  is  called 
composition,  which  is  an  addition  of  the  terms  of  a  ratio. 
(Art.  397,  2.) 

If        a  \  b\  \c  I  d 
And    h\l\\m 


d )  i2:4;:6:2) 

n)  .  io  :  5::8  :  4) 


Then     ah  :  bl\  \cm  \  dn  120  :  20!  !48  :  8 

For  from  the  nature  of  proportion,  the  two  ratios  in  the 
first  rank  are  equal,  and  also  the  ratios  in  the  second  rank. 
And  multiplying  the  corresponding  terms  is  multiplying  the 
ratios,  (Art.  361,  cor.)  that  is,  multiplying  equals  by  equals; 
(Ax.  3,)  so  that  the  ratios  will  still  be  equal,  and  therefore 
the  four  products  must  be  proportional. 

The  same  proof  is  applicable  to  any  number  of  proportions. 

*  Euclid,  11.  5.    See  Note  L. 


226  RATIO      AND      PROPORTION. 

!a  \b\\c  \  d 
h\l\\m\n 
p  :  q\  :*  :  y 

Then  ahp  I  blq\  \cmx  \  dny. 

From  this  it  is  evident,  that  if  the  terms  of  a  proportion 
be  multiplied,  each  into  itself,  that  is,  if  they  be  raised  to  any 
power,  they  will  still  be  proportional. 

if  a  :  b\\c :  d  2  :4::e  :  12 

a\b\\c\d  2  :  4!  ;6  :  12 

Then  a2  \  b2\\c2  \  d2  4  :  16!  !36  :  144 

Proportionals  will  also  be  obtained,  by  reversing  this  pro- 
cess, that  is-,  by  extracting  the  roots  of  the  terms. 

If  a  :  b\  \c  \  d,  then   <Ja  \  y/b\  \  y/c  \  y/d. 

For  taking  the  product  of  extremes  and  means,  ad=bc 
And  extracting  both  sides,  ^/ad=  y/bc 

That  is,  (Arts.  270,  383,)  y/a  \  Vb\  WC  s/<L 

Hence, 

399.  If  several  quantities  are  proportional,  their  like 
powers  or  like  roots  are  proportional* 

If  a  I  b'.  \c  \  d 

Then  an  \bn\  \cn  \  dn,  and  V«  '  Vo\  Wc  \  ^/d. 

m  m         m  nt 

And  V«n  !  1Vbn:  \l/cn  :  Vd"1,   that  is,   a"  I  b"\  \c*  \  d". 

400.  If  the  terms  in  one  rank  of  proportionals  be  divided 
by  the  corresponding  terms  in  another  rank,  the  quotients 
will  be  proportional. 

This  is  sometimes  called  the  resolution  of  ratios. 
If        a\b\\c\dr> 
And    h  \  l\\m  \  n  ) 

li     I     m     n 

This  is  merely  reversing  the  process  in  Art.  398,  and  may 
be  demonstrated  in  a  similar  manner. 

*  It  must  not  be  inferred  from  this,  that  quantities  have  the  same  ratio  m 
their  like  powers  or  like  roots.     See  Art.  358. 


12 :  g: 

:i8 : 

:  o 

6 : 2: 

:  9 : 

:  3 

12  #  6. 
6   •  2' 

.18 
•  9 

,  9 
*  3 

RATIO      AND      PROPORTION.  227 

This  should  be  distinguished  from  what  geometers  call 
division,  which  is  a  subtraction  of  the  terms  of  a  ratio. 
(Art.  397,  cor.) 

When  proportions  are  compounded  by  multiplication,  it 
will  often  be  the  case,  that  the  same  factor  will  be  found  in 
two  analogous  or  two  homologous  terms. 

Thus  if    a  '.  b\  \c  \  d*> 
And        m  \  a\  \n  \  c  > 


am  \  ab\  \cn  \  cd 

Here  a  is  in  the  two  first  terms,  and  c  in  the  two  last. 
Dividing  by  these,  (Art.  390,)  the  proportion  becomes 

m  ;  b\  \n  I  d.     Hence, 

401.  In  compounding  proportions,  equal  factors  or  divi- 
sors in  two  analogous  or  homologous  terms,  may  be  rejected. 

(a;b\\c\d  12  :    4!!9  I    3 

if  \  b :  K\d\  i  4  :  8::3  :  6 

\h\m\\l\n  8  :  20!  [6  :  15 


Then    a\m\\c\n  12  :  20!  !9  :  15 

This  rule  may  be  applied  to  the  cases,  to  which  the  terms 
"ex  aequo"  and  " ex  aequo  perturbate"  refer.  See  Arts.  393 
and  395.     One  of  the  methods  may  serve  to  verify  the  other. 

4:02.  The  changes  which  may  be  made  in  proportions, 
without  disturbing  the  equality  of  the  ratios,  are  so  numerous, 
that  they  would  become  burdensome  to  the  memory,  if  they 
were  not  reducible  to  a  few  general  principles.  They  are 
mostly  produced, 

1.  By  inverting  the  order  of  the  terms,  Art.  388. 

2.  By  multiplying  or  dividing  by  the  same  quantity,  Art.  390. 

3.  By  comparing  proportions  which  have  like  terms,  Art. 

392,  3,  4,  5. 

4.  By  adding  or  subtracting  the  terms  of  equal  ratios,  Art. 

396,  7. 

5.  By  multiplying  or  dividing  one  proportion  by  another, 

Art.  398,* 400,  1. 

6.  By  involving  or  extracting  the  roots  of  the  terms,  Art.  399. 


228  RATIO      AND      PROPORTION. 

403.  When  four  quantities  are  proportional,  if  the  first 
be  greater  than  the  second,  the  third  will  be  greater  than  the 
fourth;  if  equal,  equal:  if  less,  less. 

For,  the  ratios  of  the  two  couplets  being  the  same;  if  one 
is  a  ratio  of  equality,  the  other  is  also,  and  therefore  the  ante- 
cedent in  each  is  equal  to  its  consequent ;  (Art.  354,)  if  one 
is  a  ratio  of  greater  inequality,  the  other  is  also,  and  therefore 
the  antecedent  in  each  is  greater  than  its  consequent ;  and 
if  one  is  a  ratio  of  lesser  inequality,  the  other  is  also,  and 
therefore  the  antecedent  in  each  is  less  than  its  consequent. 

r  a=b,  c=d 
Let  a  l  b\  \c  \  d;  then  if   <  a>b,  c>d 

(  a<J),  c<df. 

Cor.  1.  If  the  first  be  greater  than  the  third,  the  second 
will  be  greater  than  the  fourth;  if  equal,  equal;  if  less,  less.* 

For  by  alternation,  a\b\\c\  d  becomes  a  \  c\  \b  \  d,  with- 
out any  alteration  of  the  quantities.  Therefore,  if  a=bf 
c=d,  &c.  as  before. 

Cor.  2.  If        a  \  m\  \c  \  n  }   .       .c         ,  7    0      . 

.     ,  7    ,         7  \  then  it  a—b,  c—d,  &c.t 

And    m  \  b.  \n  \  d  ) 

For,  by  equality  of  ratios,  (Art.  393.  2,)  or  compounding 
ratios,  (Arts.  398,  401.) 

a  :  b\  \c  :  d.     Therefore,  if  a=b,  c=d,  &c.  as  before. 

Cor.  3.  If         a  !  m\  \n  '.  d)    .        .r         7  7    0      . 

.     ,  r  then  it   a—b,  c=d,  &c.t 

And    m  \b\\c  \  n) 

For,  by  compounding  ratios,  (Arts.  398,  401,) 

a  \b\\c  \  d.     Therefore,  if  a=b,  c=d,  &c. 

4:041.  If  four  quantities  are  proportional,  their  reciprocals 
are  proportional ;  and  v.  v. 

xr  l  j      y  1111 

It  a  :  b.  ,c  :  d,  then  -  :  7 1  \-  \  -3- 
abed 

For  in  each  of  these  proportions,  we  have,  by  reduction, 
ad— be. 

*  Euclid,  14.  6,  f  Euclid,  20.  6.  \  Euclid,  21.  5. 


ratio    and    proportion.  229 

Continued    Proportion. 

4-05.  When  quantities  are  in  continued  proportion,  all 
the  ratios  are  equal.     (Art.  380.)     If 

a  \b\\b  \  c\\c  \  d\  \d  \  e, 

the  ratio  of  a  \  b  is  the  same,  as  that  of  b  \  c,  of  c  '.  d,  or  of 
d  \  e.  The  ratio  of  the  first  of  these  quantities  to  the  last,  is 
equal  to  the  product  of  all  the  intervening  ratios ;  (Art.  357,) 
that  is,  the  ratio  of  a  \  e  is  equal  to 

abed 

tX-X-vX- 

b     c     d    e 

But  as  the  intervening  ratios  are  all  equal,  instead  of  mul- 
tiplying them  into  each  other,  we  may  multiply  any  one  of 
them  into  itself ;  observing  to  make  the  number  of  factors 
equal  to  the  number  of  intervening  ratios.  Thus  the  ratio 
of  a  \  e,  in  the  example  just  given,  is  equal  to 
a  a  a  a  aA 
bXbXbX~b  =  ~b~*' 

When  several  quantities  are  in  continued  proportion,  the 
number  of  couplets,  and  of  course  the  number  of  ratios,  is 
one  less  than  the  number  of  quantities.  Thus  the  five  pro- 
portional quantities  a,  b,  c,  d,  e,  form  four  couplets  contain- 
ing four  ratios ;  and  the  ratio  of  a  :  e  is  equal  to  the  ratio  of 
a4  !  b*,  that  is,  the  ratio  of  the  fourth  power  of  the  first 
quantity,  to  the  fourth  power  of  the  second.     Hence, 

1 00.  If  three  quantities  are  proportional,  the  first  is  to  the 
third,  as  the  square  of  the  first,  to  the  square  of  the  second; 
or  as  the  square  of  the  second,  to  the  square  of  the  third.  In 
other  wTords,  the  first  has  to  the  third,  a  duplicate  ratio  of  the 
first  to  the  second.  And  conversely,  if  the  first  of  the  three 
quantities  is  to  the  third,  as  the  square  of  the  first  to  the 
square  of  the  second,  the  three  quantities  are  proportional. 
If  a  :  b\  \b  :  c,  then  a  \  c\  \a2  \  b2.     Universally, 

407.  If  several  quantities  are  in  continued  proportion, 
the  ratio  of  the  first  to  the  last  is  equal  to  one  of  the  inter- 
vening ratios  raised  to  a  power  whose  index  is  one  less  than 
the  number  of  quantities. 

If  there  are  four  proportionals,  a,  b,  c,  d,  them  a  :  d\  |<23  \  b% 
If  there  are  five,  a,b,c,d,e;  a  I  e\  '.a4  :  b*,  &c. 

20 


230  RATIO      AND      PROPORTION. 

408.  If  several  quantities  are  in  continued  proportion, 
they  will  be  proportional  when  the  order  of  the  whole  is 
inverted.  This  has  already  been  proved  with  respect  to  four 
proportional  quantities.  (Art.  388,  cor.)  It  may  be  extended 
to  any  number  of  quantities. 

Between  the  numbers,  64,  32,  16,  8,  4, 

The  ratios  are,  2,  2,  2,  2, 

Between  the  same  inverted,  4,  8,  16,  32,  64, 

The  ratios  are,  \,  \,  \,  -J-, 

So  if  the  order  of  any  proportional  quantities  be  inverted, 
the  ratios  in  one  series  will  be  the  reciprocals  of  those  in  the 
other.  For  by  the  inversion,  each  antecedent  becomes  a 
consequent,  and  v.  v.  and  the  ratio  of  a  consequent  to  its 
antecedent  is  the  reciprocal  of  the  ratio  of  the  antecedent  to 
the  consequent.  (Art.  355.)  That  the  reciprocals  of  equal 
quantities  are  themselves  equal,  is  evident  from  Ax.  4. 

4:00.  Harmonical  or  Musical  Proportion  may  be  con- 
sidered as  a  species  of  geometrical  proportion.  It  consists 
in  an  equality  of  geometrical  ratios ;  but  one  or  more  of  the 
terms  is  the  difference  between  two  quantities. 

Three  or  four  quantities  are  said  to  be  in  harmonical  pro- 
portion, when  the  first  is  to  the  last,  as  the  difference  be- 
tween the  two  first,  to  the  difference  between  the  two  last. 

If  the  three  quantities  a,  b,  and  c,  are  in  harmonical  pro- 
portion, then  a  !  c\  \a—b  \  b—c. 

If  the  four  quantities  a,  b,  c,  and  d,  are  in  harmonical 
proportion,  then  a  :  d\  \a  —  b  \  c—d. 

Thus  the  three  numbers  12,  8,  6,  are  in  harmonical  pro- 
portion. 

\nd  the  four  numbers  20,  16,  12,  10,  are  in  harmonical 
proportion. 

410.  If,  of  four  quantities  in  harmonical  proportion,  any 
three  be  given,  the  other  may  be  found.  For  from  the  pro- 
portion, 

a  I  d\  \a—b  \  c—d, 

by  taking  the  product  of  the  extremes  and  the  means,  we 
have  ac—ad=ad—bd. 

And  this  equation  may  be  reduced  so  as  to  give  the  value 
of  either  of  the  four  letters. 


RATIO      AND      PROPORTION.  231 

Thus  by  transposing  —ad,  and  dividing  by  a, 

2ad—bd 
c= . 


Examples,  in  which  the  principles  of  proportion  are  applied 
to  the  solution  of  problems. 

1.  Divide  the  number  49  into  two  such  parts,  that  the 
greater  increased  by  6,  may  be  to  the  less  diminished  by  11 ; 
as  9  to  2. 

Let  x—  the  greater,         and  49— x  =  the  less. 
By  the  conditions  proposed,  x  +  6  :  38— x\  |9  \    2 

Adding  terms,  (Art.  397,  2,)  x+6  ;  44!  19  :  11 

Dividing  the  consequents,  (Art.  390,  8,)  x+6  :  41  19  :  1 

Multiplying  the  extremes  and  means,  x+6=S6.     And  x=30. 

2.  What  number  is  that,  to  which  if  1,  5,  and  tf3,  be  seve- 
rally added,  the  first  sum  shall  be  to  the  second,  as  the  second 
to  the  third  ? 

Let  x=  the  number  required. 
By  the  conditions,  x+1  :  x-f-5  '  ]  x+5  I  x+13 

Subtracting  terms,  (Art.  397,  6,)  x  +  l  I  4!!  x+5  :  8      # 
Therefore,  8x+8=4x+20.     And  x=3. 

3.  Find  two  numbers,  the  greater  of  which  shall  be  to  the 
less,  as  their  sum  to  42 ;  arid  as  their  difference  to  6. 


Let  x  and  y- 

By  the  conditions, 

And 

By  equality  of  ratios, 

Inverting  the  means, 

Adding  and  subtracting  terms, 

Dividing  terms,  (Art.  390,) 

=  the  numbers. 

x  I  y\  \x-\-y  : 
x  :  y\  \x-y 
x+y  :  42|  \x— y 
x+y  :  x—y\  !42  \ 
(Art.  397,  7,)   2x  :  2y\  ;48  : 

x  :  y\  \  4  ; 

!  42 

:  e 
:  6 
:  6 
:  36 
:  3 

Therefore, 

3x=4y.     And  x= 

Ay 
3 

From  the  second  proportion, 

6x=yX(x- 

-y) 

4y 

Substituting  —  for  xt 

y=24.     And  x— 

32. 

232  RATIO      AND      PROPORTION. 

4.  Divide  the  number  18  into  two  such  parts,  that  the 
squares  of  those  parts  may  be  in  the  ratio   of  25  to  16. 

Let  x=  the  greater  part,  and  18— x=  the  less. 
By  the  conditions,  x2  :  (18-x)2!  |25  \  16 

Extracting,  (Art.  399,)  x  \  18— x\  ;5  S    4 

Adding  terms,  x  \  18;  !5  :    9 

Dividing  terms,  x\    2*.  *.5  :    1 

Therefore,  x  =10. 

5.  Divide  the  number  14  into  two  such  parts,  that  the 
quotient  of  the  greater  divided  by  the  less,  shall  be  to  the 
quotient  of  the  less  divided  by  the  greater,  as  16  to  9. 

Let  x=  the  greater  part,  and  14— x  =  the  less. 

x         14 — x 

By  the  conditions, 

Multiplying  terms, 
Extracting,  * 
Adding  terms, 
Dividing  terms, 
Therefore, 

6.'  If  the  number  20  be  divided  into  two  parts,  which  are 
to  each  other  in  the  duplicate  ratio  of  3  to  1,  what  number 
is  a  mean  proportional  between  those  parts  ? 

Let  x=  the  greater  part,  and  20— x=  the  less. 
By  the  conditions,  x  \  20— x\  !32  :  l2: 19  :  1 

Adding  terms,  x  \  20!  !9  \  10 

Therefore,  #=18.     And  20—^=2 

A  mean  propor.  between  18  and  2,  (Art.  384,}    =  V2X  18=6. 

7.  There  are  two  numbers  whose  product  is  24,  and  the 
difference  of  their  cubes,  is  to  the  cube  of  their  difference 
as  19  to  1.     What  are  the  numbers? 

Let  x  and  y  be  equal  to  the  two  numbers. 

1.  By  supposition,  xy=Zi  ) 

2.  And  x*—y3  :  (x— y)z\  119  :  1  5 

3.  Or,  (Art.  232,)  x*-y*  :  x*-3x2y+3xy3 -y*[  |19  :  1 

4.  Therefore,  (Art.  397,  5,)       Sx2y-3xy2  :  (x-y)3;  118  :  1 


14- 

-x  * 

x     * 

:i6 

:  o 

X2 

:<m 

-x)2\ 

:i6 

:  9 

x : 

14- 

x\  ;4 : 

3 

x : 

14: 

:4:7 

x\ 

2: 

:4 :  i 

x=8. 

i 


RATIO      AND      PROPORTION.  233 

5.  Dividing  by  x-y,  (Art.  390,  5,)         Sxy  :  (x-y)2:  !18  :  1 

6.  Or,  as  3xt/=3X  24=72,  72  :  (x-y)* \  '.18  \  1 

7.  Multiplying  extremes  and  means,  (x—y)2=4 

8.  Extracting,  x—y=^  2 

9.  By  the  first  condition,  we  have  xy=24 
Reducing  these  two  equations,  we  have     x=G,  and  y=4. 

8.  It  is  required  to  prove  that  a  \  x\  \  ^2a—y  :    ^y 
on  supposition  that                 (a+x)2  :  (a— x)2;  Ix-f-y  !  #— y. 

1.  Expanding,         a2  +2ax+x2  I  a2— 2tf#+:c2:  *.x-f-y  :  x-y 

2.  Adding  and  subtracting  terms,     2a2+2x2  :  Aax\  \2x  :  2y 

3.  Dividing  terms,  a2+x2  \  2ax\  \x  \  y 

4.  Trans,  the  factor  xy  (Art.  382,  cor.)     a2 +x2  \  2a\  \x2  \  y 

5.  Inverting  the  means,  a2  -\-x2  \  x2  \  \2a  \  y 

6.  Subtracting  terms,  a2  \  x2\  \2a—y\  \y 

7.  Extracting,  a  \  x\  \  ^2a—y  \  y/y 

9.  It  is  required  to  prove  that  dx=cy,  if  x  is  to  y  in  the 
triplicate  ratio  of  a  \  b,  and  a  \  b\  \  Vc-\-x\  \  Vd+y. 

1.  Involving*terms,  a3  :  bz\  \c-\-x  ]  d-\-y 

2.  By  the  first  supposition,  a3  \b'3\\x  \  y 

3.  By  equality  of  ratios,  c-\-x  I  d-\-y\  \x  \  y 

4.  Inverting  the  means,  c+x  \  x\  \d-\-y  \  y 

5.  Subtracting  terms,  c  \  x\\d\  y 

6.  Therefore,  dx—cy. 

10.  There  are  two  numbers  whose  product  is  135,  and  the 
difference  of  their  squares,  is  to  the  square  of  their  difference, 
as  4  to  1.     What  are  the  numbers?  Ans.  15  and  9. 

11.  What  two  numbers  are  those,  whose  difference,  sum, 
and  product,  are  as  the  numbers  2,  3,  and  5,  respectively  ? 

Ans.   10  and  2. 

12.  Divide  the  number  24  into  two  such  parts,  that  their 
product  shall  be  to  the  sum  of  their  squares,  as  3  to  10. 

Ans.   18  and  6. 

13.  Theje  are  two  numbers  which  are  to  each  other  as.  3 
to  2.  If  6  be  added  to  the  greater  and  subtracted  from  the 
less,  the  sum  and  remainder  will  be  to  each  other,  as  3  to  1. 
What  are  the  numbers?  Ans.  24  and  16. 

20* 


234  VARIATION     OR     GENERAL     PROPORTION. 

14.  There  are  two  numbers  whose  product  is  320  ;  and  the 
difference  of  their  cubes,  is  to  the  cube  of  their  difference,  as 
61  to  1.     What  are  the  numbers?  Ans.  20  and  16. 

15.  There  are  two  numbers,  which  are  to  each  other,  in 
the  duplicate  ratio  of  4  to  3 ;  and  24  is  a  mean  proportional 
between  them.  What  are  the  numbers?      Ans.  32  and  18. 

4 1 1 .  A  list  of  the  articles  in  this  section  which  contain 
the  propositions  in  the  5th  book  of  Euclid.* 

Prop. 


I. 

Art.  367.                 Prop. 

XIII. 

392,  cor. 

II. 

396. 

XIV. 

403,  cor.  1. 

III. 

390. 

XV. 

364. 

IV. 

390,  cor.  1. 

XVI. 

388. 

V. 

366. 

XVII. 

397,  cor. 

VI. 

366. 

XVIII. 

397,  2. 

VII. 

353,  cor.  1. 

XIX. 

397,  4. 

VIII. 

361,  cor.  362,  cor. 

XX. 

403,  cor.  2. 

IX. 

353,  cor.  2. 

XXI. 

403,  cor.  3. 

X. 

361,  cor.  362,  cor. 

XXII. 

394. 

XI. 

392. 

XXIII. 

395. 

XII. 

367. 

XXIV. 

396,  cor.  2. 

SECTION    XII. 


VARIATION    OR    GENERAL    PROPORTION. 

Art.  &12.  The  quantities  which  constitute  the  terms  of 
a  proportion  are,  frequently,  so  related  to  each  other,  that, 
if  one  of  them  be '  either  increased  or  diminished,  another 
depending  on  it  will  also  be  increased  or  diminished,  in  such 
a  manner,  that  the  proportion  will  still  be  preserved.  If  the 
value  of  50  yards  of  cloth  is  100  dollars,  and  the  quantity 
be  reduced  to  40  yards  ;  the  value  will,  of  course,  be  reduced 
to  80  dollars ;  if  the  quantity  be  reduced  to  30  yards,  the 
value  will  be  reduced  to  60  dollars,  &c. 

*  See  Note  M. 


ARIATION     OR     GENERAL     PROPOl 

ITION. 

yd.      yd.         dot.      dol. 

That  is,     50  :  40::  100:  80 

50  :  30::  loo:  60 

50  :  20::  loo:  40,  &c. 

235 


As  the  consequent  of  the  first  couplet  is  varied,  the  con- 
sequent of  the  second  is  varied,  in  such  a  manner,  that  the 
proportion  is  constantly  preserved. 

If  the  two  antecedents  are  A  and  B;  and  if  a  represents  a 
quantity  of  the  same  kind  with  A,  but  either  greater  or  less ; 
and  b,  a  quantity  of  the  same  kind  with  B,  but  as  many 
times  greater  or  less,  as  a  is  greater  or  less  than  A ;  then 

A  :  a::B  :b; 
that  is,  if  A  by  varying  becomes  a,  then  B  becomes  b.  This 
is  expressed  more  concisely,  by  saying  that  A  varies  as  B, 
or  A  is  as  B.  Thus  the  wages  of  a  laboring  man  vary  as 
the  time  of  his  service.  We  say  that  the  interest  of  money 
which  is  loaned  for  a  given  time,  is  proportioned  to  the  prin- 
cipal. But  a  proportion  contains  four  terms.  Here  are  only 
two,  the  interest  and  the  principal.  This  then  is  an  abridged 
statement,  in  which  two  terms  are  mentioned  instead  of  four. 
The  proportion  in  form  would  be : 

As  any  given  principal,  is  to  any  other  principal ; 

So  is  the  interest  of  the  former,  to  the  interest  of  the  latter. 

413.  In  many  mathematical  and  philosophical  investiga- 
tions, we  have  occasion  to  determine  the  general  relations  of 
certain  classes  of  quantities  to  each  other,  without  limiting 
the  inquiry  to  any  particular  values  of  those  quantities.  In 
such  cases,  it  is  frequently  sufficient  to  mention  only  two  of 
the  terms  of  a  proportion.  It  must  be  kept  in  mind,  how- 
ever, that  four  are  always  implied.  .When  it  is  said,  for 
instance,  that  the  weight  of  water  is  proportioned  to  its  bulk, 
we  are  to  understand, 

That  one  gallon,  is  to  any  number  of  gallons ; 
As  the  weight  of  one  gallon,  is  to  the  weight  of  the  given 
number  of  gallons. 

414.  The  character  cr>  is-used  to  express  the  proportion 
of  variable  quantities. 

Thus  AcnB  signifies  that  A  varies  as  B,  that,  is,  that 

A  :  a : :  B  :  b. 
The  expression  A  en  B  may  be  called  a  general  proportion. 


236        VARIATION     OR     GENERAL     PROPORTION. 

ll«i.  One  quantity  is  said  to  vary  directly  as  another, 
when  the  one  increases  as  the  other  increases,  or  is  dimin- 
ished as  the  other  is  diminished,  so  that 

A(/)Bt  that  is,  A  :  a::B  :b. 

The  interest  on  a  loan  is  increased  or  diminished,  in  pro- 
portion to  the  principal.  If  the  principal  is  doubled,  the 
interest  is  doubled ;  if  the  principal  is  trebled,  the  interest  is 
trebled,  &c. 

4 1 6.  One  quantity  is  said  to  vary  inversely  or  recipro- 
cally as  another,  when  the  one  is  proportioned  to  the  recip- 
rocal of  the  other ;  that  is,  when  the  one  is  diminished,  as 
the  other  is  increased,  so  that 

A  &  -„>  that  is,  Jl  :  «::-^:t'  or  A  :  a::b  :  B. 

In  this  case,  if  A  is  greater  than  a,  B  is  less  than  b.  (Art. 
403.)  The  time  required  for  a  man  to  raise  a  given  sum,  by 
his  labor,  is  inversely  as  his  wages.  The  higher  his  wages, 
the  less  the  time. 

417.  One  quantity  is  said  to  vary  as  two  others  jointly, 
when  the  one  is  increased  or  diminished,  as  the  product  of 
the  other  two,  so  that 

A&BC,  that  is,  A:  a:  :BC  :  be. 

The  interest  of  money  varies  as  the  product  of  the  princi- 
pal and  time.  If  the  time  be  doubled,  and  the  principal 
doubled,  the  interest  will  be  four  times  as  great. 

418.  One  quantity  is  said  to  vary  directly  as  a  second, 
and  inversely  as  a  third,  when  the  first  is  always  propor- 
tioned to  the  second  divided  by  the  third,  so  that 

B        u  ,  B      b 

A(f>-~>   that  is,  Ai  aii-~i  -- 
L>  L>     c 

410.  To  understand  the  methods  by  which  the  state- 
ments of  the  relations  of  variable  quantities  are  changed  from 
one  form  to  another,  little  more  is  necessary,  than  to  make 
an  application  of  the  principles  of  common  proportion  ;  bear- 
ing constantly  in  mind,  that  a  general  proportion  is  only  an 
abridged  expression,  in  which  two  terms  are  mentioned  instead 
of  four.  When  the  deficient  terms  are  supplied,  the  reason 
of  the  several  operations  will,  in  most  cases,  be  apparent. 


VARIATION     OR     GENERAL     PROPORTION.       237 

420.  It  is  evident,  in  the  first  place,  that  the  order  of  the 
terms  in  a  general  proportion  may  be  inverted.    (Art.  377.) 

If        A  :  a:  :B  :  b,  that  is,  if  Acn  B, 
Then  B  :b:;A:a,  that  is,      B<d  A. 

42 1 .  If  one  or  both  of  the  terms  in  a  general  proportion, 
be  multiplied  or  divided  by  a  constant  quantity,  the  propor- 
tion will  be  preserved.  wK? 

For  multiplying  or  dividing  one  or  both  of  the  terms  is  the 
same,  as  multiplying  or  dividing  analogous  terms  in  the  pro- 
portion expressed  at  length.     (Art.  390,  and  cor.  1.) 
If  A:  a::B:b,  that  is,  if    Acd  B, 

Then   m A  :  ma;:B  :  b,  that  is,      m A <r>  B, 

And     mA  :  ma:  :mB  :  mb}    that  is,      mAwmB,  &c. 

422.  If  both  the  terms  be  multiplied  or  divided  even  by 
a  variable  quantity,  the  proportion  will  be  preserved.  For 
this  is  equivalent  to  multiplying  the  two  antecedents  by  one 
quantity,  and  the  two  consequents  by  another.     (Art.  390.) 

If  A  :  a: : B  :  b,  that  is,  if  Acd  B, 

-    Then   MA  :  ma:  :MB  :  mb,  that  is,  MAu> MB,  &c. 

Cor.  1.  If  one  quantity  varies  as  another,  the  quotient  of 
the  one  divided  by  the  other  is  constant.  In  other  words,  if 
the  numerator  of  a  fraction  varies  as  the  denominator,  the 
value  remains  the  same. 

If  A  :  a:  :B  :  b,  that  is,  if  Acd  B, 
A     a      B     b 

Then  B:b::B:b:Zl:1       (Art.  123.) 
Here  the  third  and  fourth  terms  are  equal,  because  each  is 
equal  to  1.     Of  course  the  two  first  terms  are  equal;  (Art. 
403,)  so  that  if  A  be  increased  or  diminished  as  many  times 
as  B,  the  quotient  will  be  invariably  the  same. 

Cor.  2.  If  the  product  of  two  quantities  is  constant,  one 
varies  reciprocally  as  the  other. 

If  AB:  aft::  1:1,  then  ~^~:~T::~§:Tf  or  A:  a::-^:y 

Cor.  3.  Any  factor  in  one  term  of  a  general  proportion, 
may  be  transferred,  so  as  to  become  a  divisor  in  the  other ; 
and  v.  v. 

If  AcaBC,  then  dividing  by  B,  -n&C. 


238        VARIATION     OR     GENERAL     PROPORTION. 

If  A  (n  YTTi>  then  multiplying  by  C,  ACcdj^- 

4:23.  If  two  quantities  vary  respectively  as  a  third,  then 
one  of  the  two  varies  as  the  other.     (Art.  392.) 

If        A  :  a:  :B  :  b  )  .      .    ,i  (  Acd  B 
that  is,  if 


And     C:c::B*k)  "—*> x     (  Cc»B} 
Then  A:a:: (PPc,  that  is,  A<j>  C. 

424.  If  two  quantities  vary  respectively,  as  a  third,  their 
sum  or  difference  will  vary  in  the  same  manner.    (Art.  396.) 
If       A:a::B:b)  \Aci>B 

And    C:c::E:MthatlSj1..  I  C^B; 
Then  A+C  :  a+c:  :B  :  b,  that  is,  .A+CcajB, 
And    A—  C  :  a— c::B  :  6,  that  is,  A—CtnB. 
Cor.  The  addition  here  may  be  extended  to  any  number 
of  quantities  all  varying  alike.     (Art.  396,  cor.  1.) 
If  A  c/>  B,  and  C  <j)B,  and  D  </>B,  and  E  cr>B,  then 
(JL+C+D-h^c^^ 

42o.  If  the  square  of  the  sum  of  two  quantities,  varies 
as  the  square  of  their  difference :  then  the  sum  of  their  squares 
varies  as  their  product. 

If  (A+B)2«>(A-B)2;  then  A*+B*c*AB. 
For  by  the  supposition, 

(A+B)2  :  (A-B)2::(a+b)2  :  (a-b)*. 
Expanding,    adding,    and   subtracting  terms.      (Arts.  232, 
and  397,  7.) 

2A2  +2B2  :  4AB  : :  2a2  +2b2  :  4ab. 
Or,  (Art.  390.) 

A2+B2  :AB::a2-\-b2  :  ab,  that  is,  A2+B2v>AB. 

426.  The  terms  of  one  general  proportion  may  be  mul- 
tiplied or  divided  by  the  corresponding  terms  of  another. 
(Art.  398.) 

If       A:a::B:b)  ,   ^  .     .c{  Ac/>B 


[  that  is,  if  ]   ~ 


And    C:c::D:d)"^  10'    Kl  C&D; 


Then    AC: ac::BD  :  bd,  that  is,  AC  <j> BD. 

Cor.   If  two  quantities  vary  respectively  as  a  third,  the 
product  of  the  two  will  vary  as  the  square  of  the  other. 


VARIATION      OR     GENERAL     PROPORTION.       239 

If        Acr.B 


s  then  AC<j>B2. 
And    C  (dB  ) 

4=27*  If  any  quantity  vary  as  another,  any  power  or  root 
of  the  former  will  vary,  as  a  like  power,  or  root  of  the  latter. 
(Art.  399.) 

If        A  :  a  : :  B  :  b,  that  is,  if  A  cr>B, 

Then  An:  an: :  Bn:  bn,  that  is,      An<i>  Bn, 

1  J.  1         1  .  ±  1 

And    An  :  an  ::Bn  :  bn,  that  is,  An&B\ 

438.  In  compounding  general  proportions,  equal  factoid 
or  divisors,  in  the  two  terms,  may  be  rejected.    (Art.  401.) 

If       A:a::B:b^)  CAc^B 

And    B  :  b : :  C  :  c  >  that  is,  if    }  Bw  G 
And    C:c::D:</3  L  C^D 


Then  J. :  «:  :D  :  d,  that  is,  AcnD. 

Cor.  If  one  quantity  varies  as  a  second,  the  second,  as  a 
third,  the  third,  as  a  fourth,  &c.  then  the^rstf  varies  as  the  last. 
If  AwBwCwD,  then  Acr>D. 

If  A(S)B(d-^'  tnen  -4.c/>p5    that   is,    if  the   first  varies 

directly  as  the  second,  and  the  second  varies  reciprocally  as 
the  third ;  the  first  varies  reciprocally  as  the  third. 

439.  If  any  quantity  vary  as  the  product  of  two  others, 
and  if  one  of  the  latter  be  considered  constant,  the  first  wrill 
vary  as  the  other. 

If  Wei)  LB,  and  if  B  be  constant,  then  Wcr>  L. 

Here  it  must  be  observed  that  there,  are  two  conditions ; 
First,  that  W  varies  as  the  product  of  the  two  other  quanti- 
ties ;  Secondly,  that  one  of  these  quantities  B  is  constant. 

Then,  by  the  conditions,  W :  w:  :LB  :  IB ;  B  being  the  same 
in  both  terms. 

Divid.  by  the  constant  quantity  B,  W:w:  :L:l,  that  is,  WcnL. 
And  if  L  be  considered  constant,  W<j>  B. 

Thus  the  weight  of  a  board,  of  uniform  thickness  and 
density,  varies  as  its  length  ajid  breadth.  If  the  length  is 
given,  the  weight  varies  as  the  breadth.  And  if  the  breadth 
u  given,  the  weight  varies  as  the  length. 


240       VARIATION      OR     GENERAL     PROPORTION. 

Cor.  The  same  principle  may  be  extended  to  any  number 
of  quantities.  The  weight  of  a  stick  of  timber,  of  given 
density,  depends  on  the  length,  breadth,  and  thickness.  If 
the  length  is  given,  the  weight  varies  as  the  breadth  and 
thickness.  If  the  length  and  breadth  are  given,  the  weight 
varies  as  the  thickness,  &c. 

If  WoLBTi 

Then  making  L  constant,  W<n  BT, 

And  making  L  and  B  constant,  Wen  T. 

430.  On  the  other  hand,  if  one  quantity  depends  on  two 
others ;  so  that  when  the  second  is  given,  the  first  varies  as 
the  third,  and  when  the  third  is  given,  the  first  varies  as  the 
second ;  then  the  first  varies  as  the  product  of  the  other  two. 

If  the  weight  of  a  board  varies  as  the  length,  when  the 
breadth  is  given,  and  as  the  breadth  when  the  length  is  given : 
then  if  the  length  and  breadth  both  vary,  the  weight  varies 
as  their  product. 

If         Wen  L,  when  B  is  constant,  )   .       TTr     D  T 

A      A        TXT        E>  T    '  *        *  thei1    W(f)  BL' 

And     WcdB,  when  L  is  constant,  ) 

In  demonstrating  this,  we  have  to  consider,  two  variable 
values  of  W;  one,  when  L  only  varies,  and  the  other,  when 
L  and  B  both  vary.  - 

Let  w'=  the  first  of  these  variable  values, 

And  w  =  the  other; 

So  that  W  will  be  changed  to  w(,  by  the  varying  of     L] 

And  w'  will  be  farther  changed  to  w,  by  the  varying  of  B. 
Then  by  the  supposition,  Wl  w'::L  :  I,  when  B  is  constant. 
And  w'  :w  ::B  lb.  when  B  varies. 


Mult,  correspond,  terms,     Ww' :  ww' : :  BL  :  bl.     (Art.  398.) 
Dividing  by  wf,  (Art.  390,)  Wlw::BL:bl,  i.  e.   Wv>  BL. 

The  proof  may  be  extended  to  any  number  of  quantities. 

The  weight  of  a  piece  of  timber,  depends  on  its  length, 
breadth,  thickness  and  density.  If  any  three  of  these  are 
given,  the  weight  varies  as  the  other. 

This  case  must  not  be  confounded  with  that  in  Art.  423,  cor. 
In  that,  B  is  supposed  to  vary  as  A  and  as  C,  at  the  same 
time.  In  this,  B  varies  as  A,  only  when  C  is  constant,  and 
as  C,  only  when  A  is  constant.  It  can  not  therefore  vary  as 
A  and  as  C  separately,  at  the  same  time. 


ARITHMETICAL      PROGRESSION.        ^^    241 

If  one  quantity  varies  as  another,  the  former  is  equal  to  the 
product  of  the  latter  into  some  constant  quantity. 

If  A  :  B : :  a  :  b ;  then,  whatever  be  the  value  of  a,  its  ratio 
to  b  must  be  constant,  viz.  that  of  A  :  B.  Let  this  ratio  be 
that  of  m  :  1. 

Then  A  :  B::a  :  biimll.  Therefore  A—mB\  And  a=mb. 
Hence,  if  the  ratio  between  the  two  quantities  be  found 
for  any  given  value,  it  will  be  known  for  any  other  period  of 
their  increase  or  decrease.  If  the  interest  of  100  dollars  be 
to  the  principal  as  1  :  20;  the  interest  of  1000  or  10,000  will 
have  the  same  ratio  to  the  principal. 

43 1 .  Many  writers,  in  expressing  a  general  proportion,  do 
not  use  the  term  vary,  or  the  character  which  has  here  been 
put  for  it.  Instead  of  Acd  B,  they  say  simply  that  A  is  as  B. 
It  may  be  proper  to  observe,  also,  that  the  word  given  is  fre- 
quently used  to  distinguish  constant  quantities,  from  those 
which  are  variable  ;  as  well  as  to  distinguish  known  quantities 
from  those  which  are  unknown. 


**i 


SECTION    XIII. 

ARITHMETICAL    AND   GEOMETRICAL   PROGRESSION. 

Art.  432.  Quantities  which  decrease  by  a  coinmon 
difference,  as  the  numbers  10.  8,  6,  4,  2,  are  in  continued 
arithmetical  proportion.  (Art.  380.)  Such  a  series  is  also 
called  a  progression,  which  is  only  another  name  for  con- 
tinued proportion. 

It  is  evident  that  the  proportion  will  not  be  destroyed,  if 
the  order  of  the  quantities  be  inverted.  Thus  the  numbers 
2,  4,  6,  8,  10,  are  in  arithmetical  proportion. 

Quantities,  then,  are  in  arithmetical  progression,  when  they 
increase  or  decrease  by  a  common  difference. 

When  they  increase,  they  form  what  is  called  an  ascending 
series,  as  3,  5,  7,  9,  11,  &c. 

When  they  decrease,  they  form  a  descending  series,  as 
11,  9,  7,  5,  &c. 

21 


242  ^W  ARITHMETICAL      PROGRESSION. 

The  natural  numbers,  1,  2,  3,  4,  5,  6,  &c.  are  in  arithmet- 
ical progression  ascending. 

433.  From  the  definition  it  is  evident  that,  in  an  ascend- 
ing  series,  each  succeeding  term  is  found,'  by  adding  the  com- 
mon difference  to  the  preceding  term. 

If  the  first  term  is  3,  and  the  common  difference  2 ; 

The  series  is  3,  5,  7,  9,  11,  13,  &c. 

If  the  firs^t  term  is  a,  and  the  common  difference  d; 
Then  (#4- d)is  the  second  term,  a+2d+d=a+3d,  the  fourth, 

a+ct+d=a+2d  the  3d,  a+3d+d=a+4d,  the  5th,  &c. 

12  3  4  5 

And  the  series  is  a,  a-{-d.  a -{-2d,  a  +  3d,  a-\-4d,  &c. 
If  the  first  term  and  the  common  difference  are  the  same, 
the  series  becomes  more  simple.     Thus  if  a,  is  the  first  term, 
and  the  common  difference,  and  n  the  number  of  terms, 
Then     a+a=2a  is  the  second  term, 
2a-\-a—3a  the  third  term,  &c. 
And  the  series  is  a,  2a,  3a,  4a, na. 

434*  In  a  descending  series,  each  succeeding  term  is  found, 
by  subtracting  the  common  difference  from  the  preceding  term. 

If  a  is  the  first  term,  and  d  the  common  difference,  the 

12  3  4  5 

series  is  a,  a—d,  a— 2d,  a  — 3d,  a— Ad,  &c. 

Or  the  common  difference  in  this  case  may  be  considered 
as  —d,  a  negative  quantity,  by  the  addition  of  which  to  any 
preceding  term,  we  obtain  the  following  term. 

In  this  manner,  we  may  obtain  any  term,  by  continued 
addition  or  subtraction.  But  in  a  long  series,  this  process 
would  become  tedious.  There  is  a  method  much  more  expe- 
ditious.    By  attending  to  the  series 

12  3  4  5 

a,  a+d,  a-\-2d,  a-\-3d,  a+4d,  &c. 
it  will  be  seen,  that  the  number  of  times  d  is  added  to  a  is 
one  less  than  the  number  of  the  term. 

The  second  term  is  a+d,     i.  e.  a  added  to  once  d; 
The  third  is  a-\-2d,  a  added  to  twice  d\ 

The  fourth  is  a+3d,  a  added  to  thrice  d,  &c. 

So  if  the  series  be  continued, 

The  50th  term  will  be  a-f  4<K 

The  100th  term  a+99d. 


ARITHMETICAL      PROGRESSION.  243 

If  the  series  be  descending,  the  100th  term  will  be  a— 99c?. 

In  the  last  term,  the  number  of  times  d  is  added  to  a,  is 
one  less  than  the  number  of  all  the  terms.     If  then 
a=  the  first  term,  z  =  the  last,  n=  the  number  of  terms,  we 
shall  have,  in  all  cases,  z=a  +  (n—l)xd;  that  is, 

1 3o.  In  an  arithmetical  progression,  the  last  term  is  equal 
to  the  first,  +  the  product  of  the  common  difference  into  the 
number  df  terms  less  one. 

Any  other  term  may  be  found  in  the  same  way.  For  the 
series  may  be  made  to  stop  at  any  term,  and  that  may  be 
considered,  for  the  time,  as  the  last. 

Thus  the  rath  term  =  a+(m  —  l)xd. 

If  the  first  term  and  the  common  difference  are  the  same, 
z=a-\-(n—l)a=a-\-na—a,  that  is,  z=na. 

In  an  ascending  series,  the  first  term  is,  evidently,  the 
least,  and  the  last,  the  greatest.  But  in  a  descending  series, 
the  first  term  is  the  greatest,  and  the  last,  the  least. 

436.  The  equation  z=a+(n—l)d,  not  only  shows  the 
value  of  the  last  term,  but,  by  a  few  simple  reductions,  will 
enable  us  to  find  other  parts  of  the  series.  It  contains  four 
different  quantities, 

a,  the  first  term,  n,  the  number  of  terms,  and 

z,  the  last  term,  d,  the  common  difference. 

If  any  three  of  these  be  given,  the  other  may  be  found. 

1.  By  the  equation  already  found, 

z=a  +  (n—l)d=  the  last  term. 

2.  Transposing  (n— 1)6?,  (Art.  178,) 

z—(n—l) d=  a  ==  the  first  term. 

3.  Transposing  a  in  the  1st,  and  dividing  by  »— 1, 

z  —  a 

=d=  the  common  difference. 

..  4.  Transp.  a  in  the  1st,  dividing  by  d,  and  transp.  —  1, 

z  —  a 


-f  l=7i=  the  number  of  terms. 

By  the  third  equation,  may  be  found  any  number  of  arith- 
metical means,  between  two  given  numbers.  For  the  whole 
number  of  terms  consists  of  the  two  extremes,  and  all  the 


244  ARITHMETICAL      PROGRESSION. 

intermediate  terms.  If  then  m—  the  number  of  means,  m+ 
2=n,  the  whole  number  of  terms.  Substituting  m-t-2  for  n, 
in  the  third  equation,  we  have 

z  —  a 

— — -  =d,  the  common  difference. 

771  +  1 

Prob.  1.  If  the  first  term  of  an  increasing  progression  is  7, 
the  common  difference  3,  and  the  number  of  terms  9,  what  is 
the  last  term?        Ans.  z=a  +  (rf-l)d='7  +  (d-l)>&=31. 
And  the  series  is  7,  10,  13,  16,  19,  22,  25,  28,  31. 

Prob.  2.  If  the  last  term  of  an  increasing  progression  is  60, 
the  number  of  terms  12,  and  the  common  difference  5,  what 
is  the  first  term?     Ans.  a=z-(n- l)d=60- (12-  1)X 5=5. 

Prob.  3.  Find  6  arithmetical  means,  between  1  and  43. 
Ans.  The  common  difference  is  6. 
And  the  series,  1,  7,  13,  19,  25,  31,  37,  43. 

437.  There  is  another  inquiry  to  be  made  concerning  a 
series  in  arithmetical  progression.  It  is  often  necessary  to 
find  the  sum  of  all  the  terms.  This  is  called  the  summation  of 
the  series.  The  most  obvious  mode  of  obtaining  the  amount 
of  the  terms,  is  to  add  them  together.  But  the  nature  of 
progression  will  furnish  us  with  a  method  more  expeditious. 

It  is  manifest  that  the  sum  of  the  terms  will  be  the  same, 
in  whatever  order  they  are  written.  The  sum  of  the  ascend- 
ing series,  3,  5,  7,  9,  11,  is  the  same,  as  that  of  the  descend- 
ing series,  11,  9,  7,  5,  3.  The  sum  of  both  the  series,  is,  there- 
fore, twice  as  great,  as  the  sum  of  the  terms  in  one  of  them. 
There  is  an  easy  method  of  finding  this  double  sum,  and  of 
course,  the  sum  itself  which  is  the  object  of  inquiry.  Let  a 
given  series  be  written,  both  in  the  direct,  and  in  the  inverted 
order,  and  then  add  the  corresponding  terms  together. 

Take,  for  instance,  the  series  3,    5,    7,    9,  1 1 

And  the  same  inverted,  11,    9,    7,    5,    3 


The  sums  of  the  terms  will  be  14,  14,  14,  14,  14 

Take  also  the  series         a,  a+d,     a+2d,  a+3d,  a+4d 

Take  the  same  inverted,  a+4d,   a+3d,  a+2d,  a+d,     a 

The  sums  will  be         2a+4d,  2a+4d,  2a+4d,  2a+4rf,  2a+4d 
Here  we  discover  the  important  property,  that, 


ARITHMETICAL      PROGRESSION.  245 

438.  In  an  arithmetical  progression,  the  sum  of  the  ex- 
trernes  is  equal  to  the  sum  of  any  other  two  terms  equally 
distant  from  the  extremes. 

In  the  series  of  numbers  above,  the  sum  of  the  first  and 
the  last  term,  of  the  first  but  one  and  the  last  but  one,  &c.  is 
14.  And  in  the  other  series,  the  sum  of  each  pair  of  corres- 
ponding terms  is  2a -j-4d. 

To  find  the  sum  of  all  tr\e  terms  in  the  double  series,  we 
have  onry  to  observe,  that  it  is  equal  to  the  sum  of  the  ex- 
tremes repeated  as  many  times  as  there  are  terms. 

The  sum  of  14,  14,  14,  14,  14=14X5. 

And  the  sum  of  the  terms  in  the  other  double  series  is 
(2a  +  ld)X5. 

But  this  is  twice  the  sum  of  the  terms  in  the  single  series. 
If  then  we  put 

a—  the  first  term,  n=  the  number  of  terms, 

z=  the  last,  s=  the  sum  of  the  terms, 

we  shall  have  this  equation, 

a~\~z 
s=——Xn.     That  is, 

4:39.  In  an  arithmetical  progression,  the  sum  of  all  the 

terms  is  equal  to  half  the  sum  of  the  extremes  multiplied  into 
the  number  of  terms, 

Proh.  What  is  the  sum  of  the  natural  series  of  numbers 

1,  2,  3,  4,  5,  &c.  up  to  1000? 

a  +  z            1+1000 
Ans.  s=-—  Xn= - X  1000=500500. 

If  in  the  preceding  equation,  we  substitute  for  z,  its  value 
as  given  in  Art.  436,  we  have 
2a  +  (n-l)d 

In  this,  there  are  four  different  quantities,  the  first  term  of 
the  series,  the  common  difference,  the  number  of  terms,  and 
the  sum  of  the  terms ;  any  three  of  which  being  given,  the 
fourth  may  be  found,    For,  by  reducing  the  equation,  we  have, 

2s  —  dn*+dn      . 
2t  a= - >  the  first  term. 

2n  J 

21* 


246  ARITHMETICAL      PROGRESSION. 


2S""~ ~2an 

3.  d= 1  the  common  difference. 

7i2—n  w 


V(2a-d)2+8ds-2a+d     .  ,         „ 

4.  7i= — * —3 »  the  number  01  terms. 

2a . 

i£r.  1.  If  the  first  term  of  an  increasing  arithmetical  series 
is  3,  the  common  difference  2,  and  the  number  of  terms  20 ; 
what  is  the  sum  of  the  series  ?  Ans.   440. 

2.  If  100  stones  be  placed  in  a  straight  line,  at  the  distance 
of  a  yard  from  each  other ;  how  far  must  a  person  travel,  to 
bring  them  one  by  one  to  a  box  placed  at  the  distance  of  a 
yard  from  the  first  stone  ?        Ans.  5  miles  and  1300  yards. 

3.  What  is  the  sum  of  150  terms  of  the  series 

h  \  h §  I3  %  I9  &c?         Ans- 3775- 

4.  If  the  sum  of  an  arithmetical  series  is  1455,  the  least 
term  5,  and  the  number  of  terms  30 ;  what  is  the  common 
difference?  •  Ans.  3. 

5.  If  the  sum  of  an  arithmetical  series  is  567,  the  first 
term  7,  and  the  common  difference  2 ;  what  is  the  number 
of  terms?  Ans.  21. 

6.  What  is  the  sum  of  32  terms  of  the  series  1,  1^,  2,  2^,  3, 
&c.  ?  -  Ans.  280. 

7.  A  gentleman  bought  47  books,  and  gave  10  cents  for  the 
first,  30  cents  for  the  second,  50  cents  for  the  third,  &c.  What 
did  he  give  for  the  whole  ?  Ans.  220  dollars,  90  cents. 

8.  A  person  put  into  a  charity  box,  a  cent  the  first  day  of 
the  year,  two  cents  the  second  day,  three  cents  the  third  day, 
&c.  to  the  end  of  the  year.  What  was  the  whole  sum  for 
365  days?  Ans.  667  dollars,  95  cents. 

440.  In  the  series  of  odd  numbers  1,  3,  5,  7,  9,  &c.  con- 
tinued to  any  given  extent,  the  last  term  is  always  one  less 
than  twice  the  number  of  terms. 

For  z  —  a-\-(n—  \)d.  (Art.  435.)  But  in  the  proposed  series 
a=l,  and  d=2. 

The  equation,  then,  becomes   z  =  1  +  (ti  —  l)x2=2?i—  1. 

441.  In  the  series  of  odd  numbers,   1,  3,  5,  7,  9,  &c.  the 

sum  of  the  terms  is  always  equal  to  the  sauare  of  the  number 
of  terms. 


ARITHMETICAL      PROGRESSION.  247 

For  s=%(a  +  z)n.     (Art.  439.) 
But  here  a=l,  and  by  the  last  article,  z=2n  —  1. 
The  equation,  then,  becomes  s=^(l+2n— l)w=n2. 
Thus  1-1-3=4    \ 

1+3+5=9    >  the  square  of  the  number  of  terms. 
1+3  +  5+7=16  ) 

44-3.  If  there  be  two  ranks  of  quantities  in  arithmetical 
progression,  the  sums  or  differences  will  also  be  in  arithmetical 

progression. 

For  by  the  addition  or  subtraction  of  the  corresponding 
terms,  the  ratios  are  added  or  subtracted.  (Art.  349.)  And  by 
the  nature  of  progression,  all  the  ratios  in  the  series  are  equal. 
Therefore  equal  ratios  being  added  to,  or  subtracted  from, 
equal  ratios,  the  new  ratios  thence  arising  will  also  be  equal. 

To  and  from         3,    6,    9,  12,  15,  18,  2H  C  3 

Add  and  subtract  2,    4,    6,    8,  10,  12,  14  I  2 

,  >whose  ratio  is<  — 

Sums  5,  10,  15,  20,  25,  30,  35  J   5 

Differences  1,    2,    3,    4,    5,    6,    7  J  (^  1 

443.  Jf  all  the  terms  of  an  arithmetical  progression  be 
multiplied  or  divided  by  the  same  quantity,  the  products  or 
quotients  will  be  in  arithmetical  progression. 

For  by  the  multiplication  or  division  of  the  terms,  the 
ratios  are  multiplied  or  divided  ;  (Art.  348,)  that  is,  equal 
quantities  are  multiplied  or  divided  by  the  given  quantity. 
They  vvill  therefore  remain  equal. 

If  the  series  3,    5,    7,    9,  1 1,  &,c.  be  multiplied  by  4  ; 

The  prods,  will  be  12,  20,  28,  36,  44,  &c:andif  this  be  di v.  by  2; 
The  quots.  will  be    6,  10,  14,  18,  22,  &c. 

Problems  of  various  kinds,  in  arithmetical  progression,  may 
be  solved,  by  stating  the  conditions  algebraically,  and  then 
reducing  the  equations. 

Proh.  1.  Find  four  numbers  in  arithmetical  progression, 
whose  sum  shall  be  56,  and  the  sum  of  their  squares  864. 

If  x=  the  second  of  the  four  numbers, 
And  ?/=  their  common  difference : 
The  series  will  be  x— y,  xt  #+y,  x+2y. 


248  GEOMETRICAL      PROGRESSION. 

By  the  conditions,  (x— y)  +x+ (x+y)  +  (x+2y)  =  56 

And  (x-y)2  +x2  +  (x+y)2  +(x+2y)2  =864 

That  is,  4x+2y=56 

And  4x2  +4xy+6y2  =  864 

Reducing  these  equations,  we  have  #=12,  and  y=4 

The  numbers  required,  therefore,  are       8,  12,  16,  and  20. 

Prob.  2.  The  sum  of  three  numbers  in  arithmetical  pro- 
gression is  9,  and  the  sum  of  their  cubes  is  153.  What  are 
the  numbers?  Ans.   1,  3,  and  5. 

Prob.  3.  The  sum  of  three  numbers  in  arithmetical  pro- 
gression is  15;  and  the  sum  of  the  squares  of  the  two  ex- 
tremes is  58.     What  are  the  numbers  ? 

Prob.  4.  There  are  four  numbers  in  arithmetical  progres- 
sion ;  the  sum  of  the  squares  of  the  two  first  is  34 ;  and  the 
sum  of  the  squares  of  the  two  last  is  130.  What  are  the 
numbers?  Ans.    3,  5,  7,  and  9. 

Prob.  5.  A  certain  number  consists  of  three  digits,  which 
are  in  arithmetical  progression ;  and  the  number  divided  by 
the  sum  of  its  digits  is  equal  to  26 ;  but  if  198  be  added  to  it, 
the  digits  will  be  inverted.     What  is  the  number  ? 

Let  the  digits  be  equal  to  x—y,  x,  and  x-\-y,  respectively. 
Then  the  number  =100(x— y)  +  10x  +  (x+y)  =  \llx  —  99y~ 

t.      i  ,.  .  llLc— 99y  ) 

By  the  conditions,  - =26  ( 

3x  c 

And  11  \x-my+ 198=  100(#+r/)  +  10x  +  (x-y)  ) 

Therefore,      x=S,  y=l,  and  the  number  is  234. 

Prob.  6.  The  sum  of  the  squares  of  the  extremes  of  four 
numbers  in  arithmetical  progression  is  200 ;  and  the  sum  of 
the  squares  of  the  means  is  136.     What  are  the  numbers? 

Prob.  7.  There  are  four  numbers  in  arithmetical  progres- 
sion, whose  sum  is  28,  and  their  continual  product  585, 
What  are  the  numbers  ? 

Geometrical  Progression. 

444.  As  arithmetical  proportion  continued  is  arithmetical 
progression,  so  geometrical  proportion  continued  is  geomet* 
rical  progression. 

The  numbers  64,  32,  16,  8,  4,  are  in  continued  geometrical 
proportion.     (Art.  380.) 


GEOMETRICAL      PROGRESSION.  249 

In  this  series,  if  each  preceding  term  be  divided  by  the 
common  ratio,  the  quotient  will  be  the  following  term. 
^=32,  and  3^=16,  and  4£=8,  and  f=4. 

If  the  order  of  the  series  be  inverted,  the  proportion  will 
still  be  preserved ;    (Art.  408,)  and  the  common  divisor  will 
become  a  multiplier.     In  the  series 
4,  8, 16,  32, 64,  &c.  4X2=8,  and  8X2=16,  and  16X2=32,  &c. 

445.  Quantities  then  are  in  geometrical  progression, 
when  they  increase  by  a  common  multiplier,  or  decrease  by  a 
common  divisor. 

The  common  multiplier  or  divisor  is  called  the  ratio.  For 
most  purposes,  however,  it  will  be  more  simple  to  consider 
the  ratio  as  always  a  multiplier,  either  integral  or  fractional. 

In  the  series  64,  32,  16,  8,  4,  the  ratio  is  either  2  a  divisor, 
or  \  a  multiplier. 

To  investigate  the  properties  of  geometrical  progression, 
we  may  take  nearly  the  same  course,  as  in  arithmetical  pro- 
gression, observing  to  substitute  continual  multiplication  and 
division,  instead  of  addition  and  subtraction.  It  is  evident, 
in  the  first  place,  that, 

446.  In  an  ascending  geometrical  series,  each  succeeding 
term  is  found,  by  multiplying  the  ratio  into  the  preceding  term. 

If  the  first  term  is  a,  and  the  ratio  r, 
Then  aXrc=ar,  the  second  term,  ar2Xr=ar3,  the  fourth, 
arXr=ar2,  the  third,  ar3Xr=arA,  the  fifth,  <fec. 

And  the  series  is  a,  ar,  ar2,  ar3,  arA,  ar5,  &c. 

414 7 •  If  the  first  term  and  the  ratio  are  the  same,  the 
progression  is  simply  a  series  of  powers. 

If  the  first  term  and  the  ratio  are  each  equal  to  r, 
Then  rXr=r2,  the  second  term,     r3Xr—rA,  the  fourth, 
r2Xr=r3,  the  third,  rAXr=r5,  the  fifth. 

And  the  series  is  r,  r2,  r3,  r4,  r5,  r6,  &c. 

448.  In  a  descending  series,  each  succeeding  term  is 
found  by  dividing  the  preceding  term  by  the  ratio,  or  multi- 
plying by  the  fractional  ratio. 

If  the  first  term  is  ar6,  and  the  ratio  r, 

i                i            •     arQ 
the  second  term  is  >    or  ar6Xi', 

r  r 

And  the  series  is  ar*t  ar5,  ar*,  ar3,  ar2,  ar,  a,  (fee. 


250  GEOMETRICAL      PROGRESSION. 

If  the  first  term  is  a,  and  the  ratio  r, 

_,  .      .  a    a     a      0 

The  series  is  a,  ->  — ->  -71  &c.  or  a,  ar"1,  ar  2,  &,c. 

12  3  4  5  6 

By  attending  to  the  series  a,  ar,  ar2,  ar2,  ar*,  ar5,  &c.  it 
will  be  seen  that,  in  each  term,  the  exponent  of  the  power  of 
the  ratio,  is  one  less,  than  the  number  of  the  term. 

If  then  a=  the  first  term,         r=  the  ratio, 

z=  the  last,  n—  the  number  of  terms ; 

we  have  the  equation  z=arn~l,  that  is, 

4:4:9.  In  geometrical  progression,  the  last  term  is  equal  to 
the  product  of  the  first,  into  that  power  of  the  ratio  whose 
index  is  one  less  than  the  number  of  terms. 

When  the  least  term  and  the  ratio  are  the  same,  the  equa- 
tion becomes  z=rrn~l=rn.     See  Art.  447. 

4«50.  Of  the  four  quantities,  a,  z,  r,  and  n,  any  three 
being  given,  the  other  may  be  found.* 

1.  By  the  last  article, 

z=zarn~l=  the  last  term. 

2.  Dividing  by  rn~l, 

z  '       r 

n_1  =a=  the  first  term. 

3.  Dividing  the  1st  by  a,  and  extracting  the  r&ot, 

1  _J       =r=  the  ratio. 

By  the  last  equation  may  be  found  any  number  of  geomet- 
rical means,  between  two  given  numbers.  If  m=  the  num- 
ber of  means,  m+2  =  n,  the  whole  number  of  terms.  Substi- 
tuting m+2  for  n,  in  the  equation,  we  have 


(T- 


-r,  the  ratio. 


When  the  ratio  is  found,  the  means  are  obtained  by  con- 
tinued multiplication. 

Prob.  1.  Find  two  geometrical  means  between  4  and  256. 
Ans.  The  ratio  is  4,  and  the  series  is  4,  16,  64,  256. 

*  See  Note  N. 


GEOMETRICAL      PROGRESSION.  251 

Prob.  2.  Find  three  geometrical  means  between  ^  and  9. 

Ans.  \,  1,  and  3. 

451.  The  next  thing  to  be  attended  to,  is  the  rule  for 
finding  the  sum  of  all  the  terms. 

If  any  term',  in  a  geometrical  series  be  multiplied  by  the 
ratio,  the  product  will  be  the  succeeding  term.  (Art.  446.) 
Of  course,  if  each  of  the  terms  be  multiplied  by  the  ratio,  a 
new  sertes  will  be  produced,  in  which  all  the  terms  except 
the  last  will  be  the  same,  as  all  except  the  first  in  the  other 
series.  To  make  this  plain,  let  the  new  series  be  written 
under  the  other,  in  such  a  manner,  that  each  term  shall  be 
removed  one  step  to  the  right  of  that  from  which  it  is  pro- 
duced in  the  line  above. 

Take,  for  instance,  the  series,  2,  4,  8,  16,*82 

Multiplying  each  term  by  the  ratio,  we  have      4,  8,  16,  32,  64 

Here  it  will  be  seen  at  once,  that  the  four  last  terms  in  the 
upper  line  are  the  same,  as  the  four  first  in  the  lower  line. 
The  only  terms  which  are  not  in  both,  are  the  first  of  the  one 
series,  and  the  last  of  the  other.  So  that  when  we  subtract 
the  one  series,  from  the  other,  all  the  terms  except  these  two 
will  disappear,  by  balancing  each  other. 

If  the  given  series  is  a,  ar,  ar2,  ar3,.  .  .  .arn~l 

Then  mult,  by  r,  we  have        ar,  ar2,  ar3,.  . .  ,arn~l,  arn. 

Now  let  s=  the  sum  of  the  terms. 
Then,  s=a-\-ar-\-ar2  -\-ar3 , . . . .  +arn~l, 

And  mult,  by  r,         rs=       ar+ar2  +ar3, ....  +arn~l  +ar\ 

Subtracting  the  first  equation  from  the  second,    rs-s=arn-a 

And  dividing  by  (r—  1),  (Art.  124,)  5=  — ZT~* 

In  this  equation,  arn  is  the  last  term  in  the  new  series,  and 
is  therefore  the  product  of  the  ratio  into  the  last  term  in  the 
given  series. 

Therefore  s= —  >  that  is, 

r— 1 

4:«>2.  The  sum  of  a  series  in  geometrical  progression  is 
found,  by  multiplying  the  last  term  into  the  ratio,  subtracting 
the  first  term,  and  dividing  the  remainder  by  the  ratio  less  one. 


252  GEOMETRICAL      PROGRESSION. 

Proh.  1.  If  in  a  series  of  numbers  in  geometrical  progres- 
sion, the  first  term  is  6,  the  last  term  1458,  and  the  ratio  3, 
what  is  the  sum  of  all  the  terms  ? 

rz-a      3X1458-6 

Ans.  5= -  = — =2184. 

r—  1  3—1 

Prob.  2.  If  the  first  term  of  a  decreasing  geometrical  series 
is  £,  the  ratio  \,  and  the  number  of  terms  5 ;  what  is  the  sum 
of  the  series  ? 

The  last  term=«rn-1=|x(^)4  =  Ti2. 

iXt!^-£      121 

And  the  sum  of  the  terms  =  — jZ Ti —  ===  Ifi2* 

Prob.  3.  What  is  the  sum  of  the  series,  1,  3,  9,  27,  &c.  to 
12  terms  ?  Ans.  265720. 

Prob.  4.  What  is  the  sum  of  ten  terms  of  the  series  1,  f, 

174075 

*•  A'  &C'  AnS'  "59049* 

4-53.  Quantities  in  geometrical  progression  are  propor- 
tional to  their  differences. 

Let  the  series  be  a,  ar,  ar2 ,  ar3,  ar4,  &c. 
By  the  nature  of  geometrical  progression, 

a  :  ar::ar:  ar2  :  ar2  :  ar3  : :  ar3  :  ar*,  &c. 

In  each  couplet  let  the  antecedent  be  subtracted  from  the 
consequent,  according  to  Art.  397,  6. 

Then  a  :  ar:  :ar—a  :  ar2  —ar:  :ar2  —ar  :  ar3  —  ar2,  &c. 

That  is,  the  first  term  is  to  the  second,  as  the  difference 
between  the  first  and  second,  to  the  difference  between  the 
second  and  third  ;  and  as  the  difference  between  the  second 
and  third,  to  the  difference  between  the  third  and  fourth,  <fcc. 

Cor.  If  quantities  are  in  geometrical  progression,  their 
differences  are  also  in  geometrical  progression. 

Thus  the  numbers         3,     9,      27,      81,        243,  &c. 
And  their  differences,       6,     18,      54,      162,        &c.  are  in 
geometrical  progression. 

454.  Several  quantities  are  said  to  be  inharmonical  pro- 
gression, when,  of  any  three  which  are  contiguous  in  the 
series,  the  first  is  to  the  last,  as  the  difference  between  the  two 
first,  to  the  difference  between  the  two  last.     See  Art.  409. 


GEOMETRICAL      PROGRESSION.  253 

Thus  the  numbers  60,  30,  20,  15,  12,  10,  are  in  harmonica] 
progression. 

For  60:20::60-30:30-20,    And  20: 12::20-15: 15-12 
And  30:15::30-20:20-15,     And  15  :  10:  :15-12  :  12-10. 

Problems  in  geometrical  progression,  may  be  solved,  as  in 
other  parts  of  algebra,  by  the  reduction  of  equations. 

Prob.  1.  Find  three  numbers  in  geometrical  progression, 
such  that  their  sum  shall  be  14,  and  the  sum  of  their  squares  84. 

Let  the  three  numbers  be  x,  y,  and  z. 
By  the  conditions,  x  :  yiiy  :  z,    or  xz—y2  \ 

And  x+y+z  =  14  > 

And  x2+y2+z2=84:  ) 

Reducing  these  equations,  we  find  the  numbers  required 
to  be  2,  4  and  8. 

Prob.  2.  There  are  three  numbers  in  geometrical  progres- 
sion whose  product  is  64,  and  the  sum  of  their  cubes  is  584. 
What  are  the  numbers  ? 

If  x  be  the  first  term,  and  y  the  common  ratio ;  the  series 
will  be  x,  xy,  xy2 . 

By  the  conditions*       xXxyXxy2,         or  x3y3  =  64    ) 
And  x3+x3y3+x3y*  =  584:  ) 

These  equations  reducexi  give  x=2,  and  y=2. 
The  numbers  required,^nerefore,  are52,  4  and  8. 

Prob.  3.  There  are  three  n  timbers  in  geometrical  progres- 
sion :  The  sum  of  the  $rst  and  last  is  52,  and  the  square  of  the 

mean  is  100.    What  are  the  numbers?     Ans.  2,  10,  and  50. 
jr 

Prob.  4.  Of  four  numbers  in  geometrical  progression,  the 

sum  of  the  two  first  is  15,  and'tke  sum  of  the  two  last  is  60. 

What  are  the  numbers  ? 

Let  the  series  be  x,  xy,  xy2,  xy3 ;  and  the  numbers  will 
be  found  to  be  5,  10,  20,  and  40. 

Prob.  5.  A  gentleman  divided  210  dollars  among  three 
servants,  in  such  a  manner,  that  their  portions  were  in  geo- 
metrical progression ;  and  the  first  had  90  dollars  more  than 
the  last.     How  much  had  each? 

22 


INFINITES      AND      INFINITESIMALS. 

/  Jb.  6.  There  are  three  numbers  in  geometrical  progres- 
sion, the  greatest  of  which  exceeds  the  least  by  15;  and  the 
difference  of  the  squares  of  the  greatest  and  the  least,  is  to 
the  sum  of  the  squares  of  all  the  three  numbers  as  5  to  7. 
What  are  the  numbers?  Ans.   5,  10,  and  20. 

Prob.  7.  There  are  four  numbers  in  geometrical  progres- 
sion, the  second  of  which  is  less  than  the  fourth  by  24 ;  and 
the  sum  of  the  extremes  is  to  the  sum  of  the  means,  as  7  to  3. 
What  are  the  numbers  ?  Ans.   1,  3,  9,  27. 


SECTION    XIV. 

INFINITES    AJSTD    INFINITESIMALS. 

Art.  45t>.  The  word  infinite  is  used  in  different  senses. 
The  ambiguity  of  the  term  has  been  the  occasion  of  much 
perplexity.  It  has  even  led  to  the  absurd  supposition  that 
propositions  directly  contradictory  to  each  other  may  be 
mathematically  demonstrated.  These  apparent  contradic- 
tions are  owing  to  the  fact,  that  what  is  proved  of  infinity 
when  understood  in  one  particular  manjaer,  is  often  thought 
to  be  true  also,  when  the  term  has  a  very  different  significa- 
tion. Tho  two  meanings  are  insensibly  shifted,  the  one  for 
the  other,  so  that  the  proposition  which  is  really  demonstra- 
ted, is  exchanged  for  another  which  is  false  and  absurd.  To 
prevent  mistakes  of  this  nature,  it  is  important  that  the  differ- 
ent meanings  be  carefully  distinguished  from  each  other. 

456.  Infinite,  in  the  highest,  and  perhaps  the  most  proper 
sense  of  the  word,  is  that  which  is  so  great,  that  nothing  can 
be  added  to  it,  or  supposed  to  be  added. 

In  this  sense,  it  is  frequently  used  in  speaking  of  moral  and 
metaphysical  subjects.  Thus,  by  infinite  wisdom  is  meant 
that  which  will  not  admit  of  the  least  addition.  Infinite  power 
is  that  which  can  not  possibly  be  increased,  even  in  suppo- 
sition. This  meaning  of  infinity  is  not  applicable  to  the 
mathematics.  That  which  is  the  subject  of  the  mathematics 
is  quantity;  (Art.  1,)  such  quantity  ^as  may  be  conceived  of  by 
the  human  mind.     But  no  idea  can  be  formed  of  a  quantity 


INFINITES     AND     INFINITESIMALS.  255 

so  great  that  nothing  can  be  supposed  to  be  added  to  it.  In 
this  sense,  an  infinite  number  is  inconceivable.  We  may  in- 
crease a  number  by  continual  addition,  till  we  obtain  one  that 
shall  exceed  any  limits  which  we  please  to  assign.  By  this, 
however,  we  do  not  arrive  at  a  number  to  which  nothing  can 
be  added ;  but  only  at  one  that  is  beyond  any  limits  which 
we  have  hitherto  set.  Farther  additions  may  be  made  to  it 
with  the  same  ease,  as  those  by  which  it  has  already  been 
increased  so  far.  It  is  therefore  not  infinite,  in  the  sense  in 
which  the  term  has  now  been  explained.  It  is  absurd  to 
speak  of  the  greatest  possible  number.  No  number  can  he 
imagined  so  great  as  not  to  admit  of  being  made  greater. 
We  must  therefore  look  for  another  meaning  of  infinity,  before 
we  can  apply  it,  with  propriety,  to  the  mathematics. 

457.  A  mathematical  quantity  is  said  to  be  infinite,  when 
it  is  supposed  to  be  increased  beyond  any  determinate  limits. 

By  determinate  limits  are  meant  such  as  can  be  distinctly 
stated.*  In  this  sense,  the  natural  series  of  numbers,  1,  2,. 3,  4, 
5,  &c.  may  be  said  to  be  infinite.  For,  if  any  number  be  men- 
tioned ever  so  great,  another  may  be  supposed  still  greater. 

The  two  significations  of  the  word  infinite  are  liable  to  be 
confounded,  because  they  are  in  several  points  of  view  the 
same.  The  higher  meaning  includes  the  lower.  That  which 
is  so  great  as  to  admit  of  no  addition,  must  be  beyond  any 
determinate  limits.  But  the  lower  does  not  necessarily  imply 
the  higher.  Though  number  is  capable  of  being  increased 
beyond  any  specified  limits ;  it  will  not  follow,  that  a  number 
can  be  found  to  which  no  farther  additions  can  be  made. 
The  two  infinites  agree  in  this,  that  according  to  each,  the 
things  spoken  of  are  great  beyond  calculation.  But  they 
differ  widely  in  another  respect.  To  the  one,  nothing  can  be 
added.     To  the  other,  additions  can  be  made  at  pleasure. 

4rt>8.  In  the  mathematical  sense  of  the  term,  there  is  no 
absurdity  in  supposing  one  infinite  greater  than  another. 

We  may  conceive  the  numbers  "       2  2  2  2  2  2  2,  &c. 

And  4  4  4  4  4  4  4,  &c. 

to  be  each  extended,  so  far  as  to  reach  round  the  globe,  or  to 
the  most  distant  visible  star,  or  beyond  any  greater  boundary 
which  can  be  mentioned.     But  if  the  two  series  be  equally 

*  See  Note  0. 


256  INFINITES      AND     INFINITESIMALS. 

extended,  the  amount  of  the  one  will  be  twice  as  great  as  the 
other,  though  both  be  infinite. 

So  if  the  series  a+  a2+  «3+  a*  +  a5,  &c. 

And  9a+9a*+9a*+9a*+9a5,  &c. 

be  extended  together  beyond  any  specified  limits,  one  will  be 
nine  times  as  great  as  the  other.  But  it  would  be  absurd  to 
suppose  one  quantity  greater  than  another,  if  the  latter  were 
already  so  great  that  nothing  could  be  added  to  it. 

4«5!>.  An  infinite  number  of  terms  must  not  be  mistaken 
for  an  infinite  quantity.  The  terms  may  be  extended  beyond 
any  given  limits,  when  the  amount  of  the  whole  is  a  finite 
quantity,  and  even  a  small  one.  If  we  take  half  of  a  unit ; 
then  half  of  the  remainder;  half  of  the  remaining  half,  &c. 
we  shall  have  the  series 

1_U1_L14-_1 Ll_      ArP 

2^4     •     8     »     16     '     3  2>     06C" 

In  which  each  succeeding  term  is  half  of  the  preceding  one. 
Let  the  progression  be  continued  ever  so  far,  the  sum  of  all 
the  terms  can  never  exceed  a  unit.  For,  by  the  supposition, 
there  is  still  a  remainder  equal  to  the  last  term.  And  this 
remainder  must  be  added,  before  the  amount  of  the  whole 
can  be  equal  to  a  unit. 

So  f+f+f +  tV+38t+tV»  &c-  can  never  exceed  8. 
460.   When  a  quantity  is  diminished  till  it  becomes  less 
than  any  determinate  quantity,  it  is  called  an  infinitesimal. 

Thus,  in  a  series  of  fractions,  TV,  T±o,  ToV  o>  Tolo  o>  &c-  a 
unit  is  first  divided  into  ten  parts,  then  into  a  hundred,  a 
thousand,  &c.  One  of  these  parts  in  each  succeeding  term 
is  ten  times  less  than  in  the  preceding.  If  then  the  progres- 
sion be  continued,  a  portion  of  a  unit  may  be  obtained  less 
than  any  specified  quantity.  This  is  an  infinitesimal,  and  in 
mathematical  language,  is  said  to  be  infinitely  small.  By 
this,  however,  we  are  not  to  understand  that  it  can  not  be 
made  le$s.  The  same  process  that  has  reduced  it  below  any 
limit  which  we  have  yet  specified,  may  be  continued,  so  as 
to  diminish  it  still  more.  And  however  far  the  progression 
may  be  carried,  we  shall  never  arrive  at  a  point  where  we 
must  necessarily  stop. 

401.  In  the  sense  now  explained,  mathematical  quantify 
may  be  said  to  be  infinitely  divisible;  that  is,  it  may  be  sup- 
posed to  be  so  divided,  that  the  parts  shall  be  less  than  any 


INFINITES     AND      INFINITESIMALS.  257 

determinate  quantity,  and  the  number  of  parts  greater  than 
any  given  number. 

In  the  series,  TV,  tin  ToV o>  Toioo.  &c.  a  unit  is  divided 
into  a  greater  and  greater  number  of  parts,  till  they  become 
infinitesimals,  and  the  number  of  them  infinite,  that  is,  such  a 
number  as  exceeds  any  given  number.  But  this  does  not  prove 
that  we  can  ever  arrive  at  a  division  in  which  the  parts  shall 
be  the  least  possible  or  the  number  of  parts  the  greatest  possible. 

462.  One  infinitesimal  may  be  less  than  another. 
The  series  T\,  T^,  To-V,,  T<>£o  o'  &c. ) 

And  3  3  3  3  Sj-n     \ 

'rLIlu  To»    Too'    Too"o'    Toooo'    °^c*  ; 

may  be  carried  on  together,  till  the  last  term  in  each  becomes 
infinitely  small ;  and  yet  one  of  these  terms  will  be  only  half 
as  great  as  the  other.  For  the  denominators  being  the  same, 
the  fractions  will  be  as  their  numerators,  (Art.  364,  cor.  1,) 
that  is,  as  6:3,  or  2  :  1. 

Two  quantities  may  also  be  divided,  each  into  an  infinite 
number  of  parts,  using  the  term  infinite  in  the  mathematical 
sense,  and  yet  the  parts  of  one  be  more  numerous  than  those 
of  the  other. 


i ne  series   10,    100J    iooo»   ioooo'  Q^c. 
And  -1-    -i- ? --1 <W 

-t*-XiVl  4.0J      a  o  03      4.  o  o  o  3     400003     *-*-"-'• 


4  0'      400'      4000'      4  0  0  0  03     «^»  J 

may  both  be  infinitely  extended ;  and  yet  a  unit  in  the  last 
series,  is  divided  into  four  times  as  many  parts  as  in  the  first. 
But  if,  by  an  infinite  number  of  parts  were  meant,  such  a 
number  as  could  not  be  increased,  it  would  be  absurd  to  sup- 
pose the  divisions  of  any  quantity  to  be  still  more  numerous.* 

403.  For  all  practical  purposes,  an  infinitesimal  may  be 
considered  as  absolutely  nothing.  As  it  is  less  than  any 
determinate  quantity,  it  is  lost  even  in  numerical  calcula- 
tions. In  algebraic  processes,  a  term  is  often  rejected  as 
of  no  value,  because  it  is  infinitely  small. 

It  is  frequently  expedient  to  admit  into  a  calculation,  a 
small  error,  or  what  is  suspected  to  be  an  error.  It  may  be 
difficult  either  to  avoid  the  objectionable  part,  or  to  ascertain 
its  exact  value,  or  even  to  determine,  without  a  long  and 
tedious  process,  whether  it  is  really  an  error  or  not.  But  if 
it  can  be  shown  to  be  infinitely  small,  it  is  of  no  account  in 
practice,  and  may  be  retained  or  rejected  at  pleasure. 

*  See  Note  P. 
22* 


258  INFINITES      AND      INFINITESIMALS. 

It  is  impossible  to  find  a  decimal  which  shall  be  exactly 
equal  to  the  vulgar  fraction  ^.  Dividing  the  numerator  by 
the  denominator,  we  obtain  in  the  first  place  TV  This  is 
nearly  equal  to  ^.     But  j\\  is  nearer,  TVo3o  stin  nearer,  &c. 

The  error,  in  the  first  instance,  is  ¥V 

For  -3-4--1-:=_9_4-_i_=±o  =  ! 

XW1       10     '30  30     '30  30  3* 

In  the  same  manner  it  may  be  shown,  that 

the  difference  between    \  *  amj  Jf'    !§  ***     i 
( f  and  .333,  is  3  oVo>  &c- 

If  the  decimal  be  supposed  to  be  extended  beyond  any  as- 
signable limit,  the  difference  still  remaining  will  be  infinitely 
small.  As  this  error  is  less  than  any  given  quantity,  it  is  of 
no  account,  and  may  be  considered  "in  calculation  as  nothing. 

464.  From  the  preceding  example  it  will  be  seen,  that  a 
quantity  may  be  continually  coming  nearer  to  another,  and 
yet  never  reach  it.  The  decimal  0.3333333,  &c.  by  repeated 
additions  on  the  right,  may  be  made  to  approximate  continu- 
ally to  \,  but  can  never  exactly  equal  it.  A  difference  will 
always  remain,  though  it  may  become  infinitely  small. 

When  one  quantity  is  thus  made  to  approach  continually 
to  another,  without  ever  passing  it ;  the  latter  is  called  a 
limit  of  the  former.  The  fraction  §  is  a  limit  of  the  decimal 
0.666,  &c.  indefinitely  continued. 

4 6 5.  Though  an  infinitesimal  is  of  no  account  of  itself,  yet 
its  effect  on  other  quantities  is  not  always  to  be  disregarded. 

When  it  is  a  factor  or  divisor,  it  may  have  an  important 
influence.  It  is  necessary,  therefore,  to  attend  to  the  rela- 
tions which  infinites,  infinitesimals,  and  finite  quantities  have 
to  each  other.  As  an  infinitesimal  is  less  than  any  assigna- 
ble quantity,  as  it  is  next  to  nothing,  and,  in  practice,  may  be 
considered  as  nothing,  it  is  frequently  represented  by  0. 

An  infinite  quantity  is  expressed  by  the  character  00. 

460.  As  an  infinite  quantity  is  incomparably  greater  than 
a  finite  one,  the  alteration  of  the  former,  by  an  addition  or 
subtraction  of  the  latter,  may  be  disregarded  in  calculation. 
A  single  grain  of  sand  is  greater  in  comparison  with  the 
whole  earth,  than  any  finite  quantity  in  comparison  with  one 
which  is  infinite.  If  therefore  infinite  and  finite  quantities 
are  connected  by  the  sign  -f  or  — ,  the  latter  may  be  rejected 
as  of  no  comparative  value.     For  the  same  reason,  if  finite 


INFINITES      AND      INFINITESIMALS.  259 

quantities  and  infinitesimals  are  connected  by  +  or  — ,  the 
latter  may  be  expunged. 

467.  But  if  an  infinite  quantity  be  multiplied  by  one 
which  is  finite,  it  will  be  as  many  times  increased  as  any 
other  quantity  would,  by  the  same  multiplier. 

If  the  infinite  series  2  2  2  2  2  2,  &c.  be  multiplied  by  4 ; 

The  product  will  be  8  8  8  8  8  8,  &c.  four  times  as  great 
as  the  multiplicand.     See  Art.  458. 

468.  And  if  an  infinite  quantity  be  dividedby  a  finite  quan  • 
tity,  it  will  be  altered  in  the  same  manner  as  any  other  quantity. 

If  the  infinite  series  6  6  6  6  6  6  6  6,  &c.  be  divided  by  2 ; 
The  quotient  will  be  3333333  3,  &c.  half  as  great  as 
the  dividend. 

4:69.  If  a  finite  quantity  be  multiplied  by  an  infinitesi- 
mal, the  product  will  be  an  infinitesimal ;  that  is,  putting  z 
for  a  finite  quantity,  and  0  for  an  infinitesimal,  (Art.  465,) 

%X0  =  0. 

If  the  multiplier  were  a  unit,  the  product  would  be  equal 
to  the  multiplicand.  (Art.  85.)  If  the  multiplier  is  less  than 
a  unit,  the  product  is  proportionally  less.  If  then  the  multi- 
plier is  infinitely  less  than  a  unit,  the  product  must  be  infinitely 
less  than  the  multiplicand,  that  is,  it  must  be  an  infinitesimal. 
Or,  if  an  infinitesimal  be  considered  as  absolutely  nothing, 
then  the  product  of  z  into  nothing  is  nothing.     (Art.  106.) 

470.  On  the  other  hand,  if  a  finite  quantity  be  divided 
by  an  infinitesimal,  the  quotient  will  be  infinite. 

% 

o=GO- 

For,  the  less  the  divisor,  the  greater  the  quotient.  If  then 
the  divisor  be  infinitely  small,  the  quotient  will  be  infinitely 
great.  In  other  words,  an  infinitesimal  is  contained  an  infi- 
nite number  of  times  in  a  finite  quantity.  This  may,  at  first, 
appear  paradoxical.  But  it  is  evident,  that  the  quotient  must 
increase  as  the  divisor  is  diminished. 

Thus  6-f-3  =  2,  6-H0.03  =200, 

G-f-0.3=20,  6-^-0.003=2000,  &c. 

If  then  the  divisor  be  reduced,  so  as  to  become  less  than 
any  assignable  quantity,  the  quotient  must  be  greater  than 
any  assignable  quantity. 


260  INFINITES      AND      INFINITESIMALS. 

471.  If  a  finite  quantity  be  divided  by  an  infinite  quan- 
tity, the  quotient  will  be  an  infinitesimal. 

z 

For  the  greater  the  divisor,  the  less  the  quotient.  If  then, 
while  the  dividend  is  finite,  the  divisor  be  infinitely  great,  the 
quotient  will  be  infinitely  small. 

4=72.  It  is  evident  from  Art.  469,  that  -  =0,  that  is,  if  an 

infinitesimal  be  divided  by  a  finite  quantity,  the  quotient  will 
be  an  infinitesimal. 

It  must  not  be  forgotten,  that  the  expressions  infinitely  great 
and  infinitely  small,  are,  all  along,  to  be  understood  in  the 
mathematical  sense  according  to  the  definitions  in  Arts.  457, 
and  460. 

4L7 '•!•  The  expression  -  occurs  frequently,  in  the  higher 

departments  of  the  mathematics,  particularly  in  the  differen- 
tial and  integral  calculus  ;  which  treat  principally  of  variable 
quantities,  and  in  which,  these  are  often  considered  as  re- 
duced to  infinitesimals.  Even  in  this  state,  there  may  be  a 
definite  ratio  between  them.     This,  however,  is  not  indicated 

by  the  expression  -•     If   all  infinitesimals  were  equal,   the 

quotient  of  one  divided  by  another  would  invariably  be  unity. 
But  as  one  of  them  may  be  greater  than  another,  (Art.  458,) 

the  numerator  of  the  fraction  -  may  be  greater  or  less  than 

the  denominator.  The  value  of  the  expression  taken  ab- 
stractly, without  reference  to  its  origin  in  particular  cases, 
is  said  to  be  indeterminate.  But  when  we  trace  it  back,  to 
the  quantities  of  assignable  magnitude  from  which  it  has  been 
derived,  we  may  discover  the  ratio  which  the  dividend  bears 
to  the  divisor :  to  take  a  very  simple  example, 

a2-b2 

— =a  +  fc.     (Art.  130.) 

a—b  N 

Considering  b  as  a  variable  quantity,  if  it  be  made  equal 

0 
to  a,  we  have  -  =2a. 


INFINITES      AND      INFINITESIMALS.  261 

4:7 4=*  The  expression  -  is  frequently  derived  from  a  frac- 
tion which  has  a  common  factor  in  the  numerator  and  de- 
nominator. When  this,  on  a  particular  supposition,  becomes 
0,  it  reduces  to  0  the  expressions  into  which  it  entered  as  a 
factor.  (Art.  469.)  If  it  be  removed  by  division,  the  definite 
ratio  of  the  numerator  of  the  fraction  to  the  denominator 
may  be  ascertained.  Thus  in  the  example  just  given,  if  we 
suppose  b=a,  (a2 —b2)  and  a— b  are  each  reduced  to  0. 
But  dividing  by  the  common  factor  (a—b)  we  have  a+b, 
or  2a,  as  before. 

Though  dividing  the  terms  of  a  fraction  by  a  common 

factor  does  not  alter  the  ratio  between  them,  (Art.  145,)  yet 

it  may  enable  us  to  discover  the  ratio  which  still  continues, 

.      .  .       0 

even  in  the  expression  -« 

a3  —  b*  0 

Ex.  2.  The  fraction  — — r-  becomes  ->  when  b  is  made 
a2  —  b2  0 

equal  to  a.     But  if  it  be  divided  by  the  common  factor  a— & 

•              a2+ab+b2 
(Art.  130,)  the  quotient  is    —7 »    which,  when    b=a 

.  Sa2      Sa        _  n   . 

becomes   — —  =  — ->  a  definite  quantity. 

--       •     mi      r       .       x*+ax5  —  9a2x2  +  lla3x—4aA    _ 

Ex.  3.    The  fraction  — —  — — — -    be- 

x4—  a#3— 3a2x2+  5a3x— 2a4 

comes  -  when  #=#.     But  if  the  numerator  and  denomina- 
tor be  divided  by  the  common  factor  (x—a)3,  the  fraction 

x~\~\a  5 

becomes    — — - -  which  =-  when  x=a. 
x+2a  3 

475.   In  other  cases,  a  fraction  which,  on  a  particular 

supposition,  gives  ->  when  reduced  to  its  lowest  terms,  may 

retain  the  0,  either  in  the  numerator  or  the  denominator ;  so 
that  its  value  will  be  either  0  or  oo. 

r?      a    tu    r      ♦•        Za3-\0a2x+4ax2+8x3  .  0 

Ax.  4.    1  he  traction becomes  -> 

a2—  4x2  0 

on  the  supposition  that  x=\a.     But  if  it  be  divided  by  the 


262  COMMON      MEASURE. 

2a2  —4ax—4x2 

common  factor  a— 2x  which  =0,  it  becomes  r— 

a-\-2x 

which  =  — - »  an  infinitesimal,  (Art.  472.) 

L.  mt      „  3a2—4ax—4x2  .  0 

Ex.  5.  The  fraction  — — — — _  ,  Q   ,     becomes     - 

a3— 2a2x— 4ax2+8x2  0 

when  x—\a.     If  it  be  divided  by  the  common  factor  a— 2xt  it 

3a+2x      _  .  _       Sx    "     .    .   . 
becomes    a  2  which  =-—  >  an  infinite  quantity,  (Art. 470.) 


SECTION    XV. 


COMMON  MEASURE,  AND  MULTIPLE,  PERMUTA- 
TIONS AND  COMBINATIONS. 

Art.  476.  The  Greatest  Common  Measure  of  two  quan- 
tities may  be  found  by  the  following  rule ; 

Divide  one  of  the  quantities  by  the  other,  and  the  preceding 
divisor  by  the  last  remainder,  till  nothing  remains  ;  the  last 
divisor  will  be  the  greatest  common  measure. 

The  algebraic  letters  are  here  supposed  to  stand  for  whole 
numbers.  In  the  demonstration  of  the  rule,  the  following 
principles  must  be  admitted. 

1.  Any  quantity  measures  itself,  the  quotient  being  1. 

2.  If  two  quantities  are  respectively  measured  by  a  third, 
their  sum  or  difference  is  measured  by  that  third  quantity. 
If  b  and  c  are  each  measured  by  d,  it  is  evident  that  b+c, 
and  b—c  are  measured  by  d.  Connecting  them  by  the  sign 
+  or  — ,  does  not  affect  their  capacity  of  being  measured  by  d. 

Hence,  if  b  is  measured  by  d,  then  by  the  preceding  propo- 
sition, b+d  is  measured  by  d. 

3.  If  one  quantity  is  measured  by  another,  any  multiple 
of  the  former  is  measured  by  the  latter.  If  b  is  measured 
by  d,  it  is  evident  that  b-\  b  3b,  4b,  nb,  &c.  are  measured 
by  d. 


COMMON      MEASURE.  263 

Now  let  D=  the  greater,  and  d=  the  less  of  two  algebraic 
quantities,  whether  simple  or  compound.  And  let  the  pro- 
cess of  dividing,  according  to  the  rule  be  as  follows : 

d)D(q 

r)d(q> 
rq' 


r')r{q" 
r'q" 


In  which  q,  q' ',  q",  are  the  quotients,  from  the  successive 
divisions ;  and  r,  r',  and  o  the  remainders.  And  as  the  divi- 
dend is  equal  to  the  product  of  the  divisor  and  quotient  added 
to  the  remainder, 

D—dq-\-r,  and  d—rq'-\-r'. 

Then,  as  the  last  divisor  r'  measures  r  the  remainder  being  o, 
it  measures  (2,  and  3,)    rq'+r'=d, 
and  measures  dq-\-r=D. 

That  is,  the  last  divisor  r'  is  a  common  measure  of  the 
two  given  quantities  D  and  d. 

It  is  also  their  greatest  common  measure.  For  every  com- 
mon measure  of  D  and  d,  is  also  (3,  and  2,)  a  measure  of 
D—dq=r;  and  every  common  measure  of  d  and  r,  is  also  a 
measure  of'd-rqf=r'.  But  the  greatest  measure  of  r'  is  itself. 
This,  then,  is  the  greatest  common  measure  of  D  and  d. 

The  demonstration  will  be  substantially  the  same,  what- 
ever be  the  number  of  successive  divisions,  if  the  operation 
be  continued  till  the  remainder  is  nothing. 

To  find  the  greatest  common  measure  of  three  quantities ; 
first  find  the  greatest  common  measure  of  two  of  them,  and 
then,  the  greatest  common  measure  of  this  and  the  third 
quantity.  H  the  greatest  common  measure  of  D  and  d  be 
r',  the  greatest  common  measure  of  r'  and  c,  is  the  greatest 
common  measure  of  the  three  quantities  D,  d,  and  c.  For 
every  measure  of  r',  is  a  measure  of  D  and  d;  therefore  the 
greatest  common  measure  of  r'  and  c,  is  also  the  greatest 
common  measure  of  D,  d,  and  c. 

The  rule  may  be  extended  to  any  number  of  quantities. 


264  COMMON      MEASURE. 

477.  There  is  not  much  occasion  for  the  preceding  ope- 
rations, in  finding  the  greatest  common  measure  of  simple 
algebraic  quantities.  For  this  purpose,  a  glance  of  the  eye 
will  generally  be  sufficient.  In  the  application  of  the  rule  to 
compound  quantities,  it  will  frequently  be  expedient  to  reduce 
the  divisor,  or  enlarge  the  dividend,  in  conformity  with  the 
following  principle : 

The  greatest  common  measure  of  two  quantities  is  not 
altered,  by  multiplying  or  dividing  either  of  them  by  any 
quantity  which  is  not  a  divisor  of  the  other,  and  which  con- 
tains no  factor  which  is  a  divisor  of  the  other. 

The  common  measure  of  ab  and  ac  is  a.  If  either  be 
multiplied  by  d,  the  common  measure  of  abd,  and  ac,  or  of 
ab  and  acd,  is  still  a.  On  the  other  hand,  if  ab  and  acd  are 
the  given  quantities,  the  common  measure  is  a ;  and  if  acd 
be  divided  by  d,  the  common  measure  of  ab  and  ac  is  a. 

Hence  in  finding  the  common  measure  by  division,  the 
divisor  may  often  be  rendered  more  simple,  by  dividing  it  by 
some  quantity,  which  does  not  contain  a  divisor  of  the  divi- 
dend. Or  the  dividend  may  be  multijAied  by  a  factor,  which 
does  not  contain  a  measure  of  the  divisor.  This  may  be 
necessary,  when  the  first  term  of  the  divisor  is  not  contained 
in  the  first  term  of  the  dividend. 

Ex.  1.  Find  the  greatest  common  measure  of 

Ga2  +  llax+3x2,  and  Ga2  +*7ax-3x2. 
6a2  +7ax-3x2)Ga2  +  llax+3x2  (1 
6a2  +  lax—3x2 


Dividing  by      2x)4ax+Gx2 

2a+ 3x)Ga2  +7ax-3x2  (Sa—x 
Ga2  +9ax 


—2ax—3x2 
—2ax—3x2 


Alter  the  first  division  here,  the  remainder  is  divided  by 
2x,  which  reduces  it  to  2a -f  3x.  The  division  of  the  pre- 
ceding divisor  by  this,  leaves  no  remainder.  Therefore 
2a+3x  is  the  common  measure  required. 


COMMON      MEASURE.  265 

2.  What  is  the  greatest  common  measure  of  x3  —  b2x,  and 
x2+2bx+b2  ?  Ans.  x+b. 

3.  What  is  the  greatest  common  measure  of  cx+x2,  and 
a2c+a2x?  Ans.  c+x. 

4.  What  is  the  greatest  common  measure  of  3x3-24x-9, 
and  2x3-16x—6l  Ans.   x3  —  8x— 3. 

5.  What  is  the  greatest  common  measure  of  a4—  b*,  and 
a5-b2a3?  Ans.   a2 -b2 . 

6.  What  is  the  greatest  common  measure  of  x2  —  1,  and 
xy+y?      •  Ans.   #  +  1. 

7.  What  is  the  greatest  common  measure  of  #3  — a3,  and 
.T4-a4? 

8.  What  is  the  greatest  common  measure  of  a2  —ab—2b*i 
and  a2-Sab+2b2  ? 

9.  What  is  the  greatest  common  measure  of  a4— z4,  and 
a3  —  a2  x  —  ax2  -\-x3  ? 

10.  What  is  the  greatest  common  measure  of  a3—  ab2, 
and  a2+2ab+b2  ? 

Least  Common  Multiple. 

478.  To  find  the  least  common  multiple  of  two  or  more 
quantities, 

Let  m=  the  least  common  multiple  of  a  and  b. 

x=  their  greatest  common  measure. 

p=  the  number  of  times  a  is  contained  as  a  factor  in  m. 

q=  the  number  of  times  b  is  contained  in  m. 
Then  ap—m,  and  bq—m.         Therefore,  ap  =  bq. 

Dividing  both  by  bp,  we  have    t  =  ~  *     And  -  is  the  frac- 

ct  o' 

tion  7  in  its  lowest  terms.     If  not,  let   —  be  this  fraction 
b  p' 

in  its  lowest  terms,  p'   and  q'   being  less  than   p   and  q. 

q'  a 
Then  as  —  =  t>  ap'=bq',  (Art.  136.)  And  as  ap'  is  divisi- 
ble by  «,  its  equal  6^'  must  also  be  divisible  by  a,  as  well  as 
by  b.  Therefore  bq'  is  a  common  multiple  of  a  and  &,  a 
multiple  less  than  bq,  or  its  equal  m ;  that  is,  less  than  the 
least  common  multiple  of  a  and  b,  which  is  impossible. 

23 


266  PERMUTATIONS. 

Now  the  number  which,  by  division,  reduces  a  fraction  to  its 
lowest  terms,  is  the  greatest  common  measure  of  the  numerator 

*  a  b 

and  denominator.  (Art.  149.)  We  have  then  -  =q,  and  -==p. 

x  x 

b  a^ 

As  m—ap,  substituting  -  for  p,  we  have,  m— — 

That  is,  The  least  common  multiple  of  two  numbers  is  equal 
to  their  product  divided  by  their  greatest  common  measure. 

Permutations. 

470.  The  different  results  obtained,  by  varying  the  order 
in  which  a  given  number  of  letters  or  things  may  be  written 
or  placed,  one  after  another,  are  called  permutations. 

Thus  ab  and  ba  are  permutation's  of  the  letters  a  and  b. 
And  abc,  acb,  bac,  bca,  cab,  and  cba,  are  permutations  of  a, 
b,  and  c ;  each  of  the  three  letters  being  on  the  left,  in  the 
middle,  and  on  the  right,  of  two  results. 

The  number  of  the  permutations  of  two  letters  is  evidently 
two,  ab  and  ba. 

To  find  the  number  of  permutations  of  three  letters,  a,  b, 
and  c;  observe 

That  a,  placed  before  each  of  the  permutations  of  b  and  c 

gives  abc,  acb 

b,  before  those  of  a  and  c  gives  bac,  bca 

c,  before  those  of  a  and  b  gives  cab,  cba 

The  whole  number  is  3X2,  =  the  whole  number  of  letters, 
multiplied  by  the  permutations  of  the  next  less  number. 

Or  thus,  In  each  of  the  permutations  of  a  and  b,  the  ad- 
ditional letter  c  may  have  three  positions,  before,  between, 
and  after,  the  other  two.  Therefore  all  the  permutations  of 
♦hree  letters  =3x2,  as  before. 

If  there  be  four  letters,  abed,  each  of  these  may  be  placed 
successively  before  the  six  permutations  of  three  letters  ; 
making  the  whole  number  from  the  four  letters,  4X3X2=24. 

Or,  the  additional  letter  d  may  have  four  positions  in  each 
of  the  permutations  of  the  other  three.  It  may  stand  before 
each  of  the  three,  and  after  them  all. 

So  if  n  be  any  number  of  letters,  and  Q  the  permutations 
of  n  —  1  letters,  the  permutations  of  n  letters  =nxQ.  For 
each  of  the  n  letters  may  be  placed  before  the  several  per- 
mutations of  7i  —  1  letters. 


PERMUTATIONS. 


267 


==  24 
5       =120 

up  to  n. 


In  this  way,  we  pass  from  the  permutations  of  a  given 
number  of  letters,  to  those  of  the  next  greater  number.  It 
follows,  that  the  number  of  permutations  corresponding  to 
each  of  the  natural  series  of  numbers  1,  2,  3,  4,  &c.  =  the 
product  of  the  factors  1.2.3.4,  &c. 

{of  three  letters  are  1.2.3  =6 

of  four  are  1.2.3. 

of>e  are  1.2.3. 

of  n  letters       are  1.2.3. 
Or,  reversing  the  order,  n(n—  l)(n— 2)(n  —  3) . .  .3  .  2  .  1. 

480.  In  the  preceding  cases,  each  permutation,  when  the 
solution  is  completed,  contains  all  the  letters  given,  in  the 
problem  to  be  solved.  But  we  have  frequent  occas'on  to 
determine  how  many  pairs,  triplets,  or  larger  sets,  can  be 
taken  from  a  greater  number  of  letters  or  things ;  for  in- 
stance, how  many  permutations  can  be  made  from  four  let- 
ters, a,  b,  c,  and  d,  taking  them  two  and  two,  ab,  ac,  ad,  &e. 
or  three  and  three,  abc,  abd,  bed,  &c. 

These  classes  of  results  are  sometimes  termed  arrange- 
ments,  to  distinguish  them  from  the  permutations  which  con- 
tain all  the  given  letters. 

Letters  taken  singly,  or  one  by  one,  are  also  called  arrange- 
ments, or  permutations. 

To  find  the  number  of  permutations  of  four  letters,  taken 
in  couplets,  or  sets  of  two  each. 

Of  the  four  letters,  abed,  three  at  a  time  may  be  re- 
served,  to  be  annexed  to  the  remaining  one. 

rb,  c,  and  d  to  a" 
a,  c,  and  d  to  b 
a,  b,  and  d  to  c 
b,  and  c  to  d 

The  number  of  couplets  =4X3  =  12=  the  number  of  single 
letters  multiplied  by  the  three  which  are  reserved  at  a  time- 
To  find  the  number  of  permutations  of  four  letters  taken 
in  triplets,  or  sets  of  three  each. 

First  find,  as  above,  the  number  of  couplets  formed  from 
the  four  letters.  As  each  one  of  these  sets,  taken  by  itsel£ 
contains  only  two  of  the  four  given  letters,  the  other  two  are, 
of  course,  excluded  from  it ;  and  are  said  to  be  reserved,  to 


Annexing 


we  have 


ab 

ac 

ad 

ba 

be 

bd 

ca 

cb 

cd 

da 

db 

dc 

268  PERMUTATIONS. 

be  annexed  to  the  couplet,  in  forming  triplets.  In  this  way, 
we  avoid  repeating  any  letter,  in  any  one  of  the  sets. 

If  to  ab,  the  first  of  the  couplets  above,  there  be  annexed 
successively  the  two  letters  which  are  not  contained  in  it, 
we  have  the  triplets  abc  and  abd.  If  to  ac,  the  second 
couplet,  there  be  annexed  the  two  letters  which  are  not  con- 
tained in  that,  we  have  acb  and  acd. 

Proceeding  in  this  manner  with  the  other  sets,  we  obtain 
twelve  pairs  of  triplets,  each  difFering  from  the  others,  either 
in  some  of  the  letters,  or  in  the  order  in  which  they  stand. 
The  whole  number  of  triplets  is  equal  to  the  product  of  the 
number  of  couplets  into  the  number  of  letters  reserved  ; 
12.2=4.3.2. 

481.  To  apply  this  mode  of  calculation  to  any  number 
of  letters  or  things ; 

Let        n—  the  whole  number  to  be  arranged. 
And        r—  the  number  required  to  be  taken  in  a  set. 
Then  will  r-\—  the  number  in  the  next  less  set 

And  n— j(r— 1)  or  n  —  r+l  =  the  difference  between  the 
whole  number  of  letters,  and  the  number  in  the  lesser  set. 
This  difference  then  is  equal  to  the  number  of  letters  reserved 
from  one  of  the  lesser  sets. 

To  obtain  then  the  number  of  permutations  from  n  letters 
a  b  c ...  to  /,  taken  in  sets  of  r  and  r,  supposing  the  per- 
mutations taken  in  sets  of  r—  1  to  be  already  known ; 

Let  P=  the  permutations  of  n  letters  in  sets  of  r—  1. 

Then  to  each  of  these,  the  reserved  letters,  whose  number 
=n— r-fl,  may  be  applied.  Therefore,  the  whole  number 
of  permutations  of  n  letters,  in  sets  of  r  and  r,  is 

Px(n-r+l) 
That  is,  It  is  equal  to  the  product  of  the  number  of  permu- 
tations in  sets  of  r—  1,  into  the  number  of  letters  reserved. 

4LS2.  To  obtain  the  number  of  permutations  in  succes- 
sion, as  we  pass  from  the  smaller  sets  to  the  larger,  observe  that 
If  r=l,  then  n  fn 


If  r=2, 

P=n 

71-1 

If  r=3, 

P=n(n—l)              And  7i— r+l  =  < 

71  —  2 

If  r=4, 

P=zn(n-l)(n-2) 

71-3 

If  r=5, 

P=n(n-l)(n-2)(n-3) 

^7i  — 4 

COMBINATIONS.  269 

Bearing  then  in  mind  that,  when  the  letters  are   taken 
singly,  the  permutations  are  equal  to  the  number  of  letters, 
the  permutations  of  n  letters,  being  equal  to 
Px(n-r+l) 
If  r=l,    are    n  , 

If  r=2,  n{n-\) 

If  r=3,  n(n-l)(n-2) 

If  r=4,  w(n-l)(7i-2)(n-3) 

If  r=5,  n(n-l)(n-2)(n--3)(n-4).  % 

From  the  manner  in  which  the  larger  of  these  formulas 
are  derived  from  the  smaller,  and  from  the  fact  that,  in  each, 
the  first  factor  is  n,  and  the  others  regularly  decreasing  by  1, 
till  the  last  is  equal  to  n— r  + 1,  it  is  evident  that,  for  still 
larger  sets,  the  general  formula  is 

7l(7l-l)(7l  — 2)(n-3)  . . .  (n-r+1). 
If  r=n,  that  is,  if  the  permutations  are  in  sets  equal  to  th» 
whole  number  of  letters,  then  n  —  r+1  becomes  1,  and  the 
formula  is 

n(n—  l)(7i— 2)(7i  — 3)...3;2.1,  the  same  as  in  Art.  479. 

Ex.  1.  How  many  permutations  can  be  made  of  the  letters 
in  the  word  chemistry  ? 

Ans.   1.2.3.4.5.6.7.8.  9=362880. 

2.  How  many  changes  can  be  rung  with  ten  bells  ? 

3.  How  many  arrangements  can  be  made  with  7  letters, 
taken  4  and  4  together  ?  Ans.   7.6.5.  4  =  840. 

4.  How  many  arrangements  can  be  made  with  15  letters, 
taken  6  and  6  ? 

Combinations. 

483,  A  combination,  in  algebra,  is  a  collection  of  letters 
or  things,  taken  without  regard  to  the  order  in  which  they 
are  placed.  Combinations  differ  from  each  other,  in  at  least 
one  of  the  letters  of  which  they  are  composed.  A  given  set 
of  letters  constitutes  only  one  combination,  however  various 
may  be  the  order  in  which  they  are  arranged.  But  a  single 
combination  contains  as  many  permutations,  as  the  changes 
which  it  admits  in  the  order  of  its  letters.  Thus  the  six  per- 
mutations abc  acb  bac  bca  cab  cba  are  all  composed  of  the 
same  letters,  differing  in  their  arrangement  only. 

23* 


270  COMBINATIONS. 

Where  there  are  several  combinations,  it  is  evident  that 
the  whole  number  of  permutations  of  n  letters,  is  equal  to 
the  number  of  combinations,  multiplied  into  the  number  of 
permutations  in  each  of  the  combinations.  Therefore,  the 
number  of  combinations  of  n  letters  is  equal  to  the  whole 
number  of  permutations,  divided  by  the  number  of  permuta- 
tions in  one  combination. 

To  find,  then,  the  number  of  combinations  of  n  letters, 
taken  in  sets  of  r  letters  at  a  time, 

Letf7= the  number  of  combinations  of  n  letters  in  sets  of  r  and  r. 
X=the  number  of  permutations  of  n  letters  in  sets  of  r  andr. 
]T=the  number  of  permutations  of  r  letters. 

Then  X=CxY.  Or,   C=~. 

Let  P=  the  number  of  permutations  of  n  letters  in  sets  of  r- 1 
as  in  Art.  481. 
Q=the  number  of  permutations  of  r— 1  letters,  as  in 
Art.  479. 
P 
Then  g  =  the  number  of  combinations  in  sets  of  r— 1. 

By  Art.  482,  X=.P(n-r+l).     mu       .        „     P    w-r+1 
By  Art.  479,   Y=QXr.  Therefore,  C=^X——  • 

To  expand  this  last  formula,  passing  from  the  smaller  num- 
bers to  the  larger,  take  the  values  of  P  for  successive  num- 
bers from  the  list  in  Art.  482,  and  the  corresponding  values 
of  Q  from  Art.  479,  and  we  have, 

If  r=2,  P=n  ^nd  if  r=2,  Q=l 

If  r=3,  P=n(n-1)  3,  Q=1.2 

If  r=4,  P=n(n-l)(n-2)  4,   Q=1.2.3 

If  r=5,  P=w(7i-l)(n-2)(w-3)  5,   Q=1.2.3.4 

p    n r+1 

Therefore,  the  formula  ^-x becomes  as  follows, 

rr  ~     71(71—1) 

Tf  r-i    *(»-l)(»-3) 


If  r=4, 
If  r=5, 


INVOLUTION      OF      BINOMIALS.  271 

ft  (n—  l)(n— 2)(n— 3) 


1.    2.         3.         4. 

7i  (ti—  1)(ti  —  2)(ti  —  3)(n— 4) 
1.    2.        3^         4^         57" 


71  —  7*-f- 1 

As  the  right  hand  factor,  in  each  case,  = »  the  fol- 
lowing is  the  general  formula  for  the  numbei  of  combinations 
of  ti  letters  taken  r  and  r. 

ft(7l-l)(7i-2)(7l-3)  7i-r+l  * 

1.    2.         JT         47" ' '  *        r 

Ex.  1.  What  is  the  number  of  combinations  of  10  letters 

i J«  a         10.9.8.7 

taken  4  and  4?  Ans.  — — —-^—=210. 

2.  What  is  the  number  of  combinations  of  9  letters  taken 
3  and  3.  a   o    y 


SECTION    XVI. 


INVOLUTION    AND    EXPANSION    OF    BINOMIALS. 

Art.  484.  The  manner  in  which  a  binomial,  as  well  as 
any  other  compound  quantity,  may  be  involved  by  repeated 
multiplications,  has  been  shown  in  the  section  on  powers. 
(Art.  230.)  But  when  a  high  power  is  required,  the  opera- 
tion becomes  long  and  tedious. 

This  has  led  mathematicians  to  seek  for  some  general  prin- 
ciple, by  which  the  involution  may  be  more  easily  and  expe- 
ditiously performed.  We  are  chiefly  indebted  to  Sir  Isaac 
Newton  for  the  method  which  is  now  in  common  use.  It  is 
founded  on  what  is  called  the  Binomial  Theorem,  the  inven- 
tion of  which  was  deemed  of  such  importance  to  mathemati- 
cal investigation,  that  it  is  engraved  on  his  monument  in 
Westminster  Abbey. 

485.  If  the  binomial  root  be  a+b>  we  may  obtain,  by 
multiplication,  the  following  powers.     (Art.  230.) 


272  INVOLUTION      OP      BINOMIALS. 

(a+b)*=a*+2ab+b* 

(a+b)*=a*  +Za2b+Zab2  +b* 

(a+by=a*+4a3b+6a2b2+4ab*+b* 

(a+b)5  =a5  +5a*b+10a3b2  +  10a2b3 +5ab*  +b5 ,  &c. 
By  attending  to  this  series  of  powers,  we  shall  find,  that 
the  exponents  preserve  an  invariable  order  through  the  whole. 
This  will  be  very  obvious,  if  we  take  the  exponents  by  them- 
selves, unconnected  with  the  letters  to  which  they  belong. 

T  ,  \  of  a  are  2,  1,  0 

In  the  square,  the  exponents        j  of  ft  ^  Q  %  g 

(  of  a  are  3,  2,  1,  0 
In  the  cube,  the  exponents  j  of  &  ^  Q  ^  ^  % 

t    ,u    in  (U  .    j  of  a  are  4,  3,  2,  1,  0 

In  the  4th  power,  the  exponents  j  of  ft  ^  Q  j  ^  g  4 

&c. 

Here  it  will  be  seen  at  once,  that  the  exponents  of  a  in 
the  first  term,  and  of  b  in  the  /<zs£,  are  each  equal  to  the  index 
of  the  power ;  and  that  the  sum  of  the  exponents  of  the  two 
letters  is  in  every  term  the  same. 

It  is  farther  to  be  observed  that  the  exponents  of  a  regu- 
larly decrease  to  0,  and  that  the  exponents  of  b  increase  from 
0.  That  this  will  universally  be  the  case,  to  whatever  extent 
the  involution  may  be  carried,  will  be  evident,  if  we  consider, 
that  in  raising  from  any  power  to  the  next,  each  term  is  mul- 
tiplied both  by  a  and  by  b. 

If  the  exponents,  before  the  multiplication,  increase  and 
decrease  by  1,  and  if  the  multiplication  adds  1  to  each,  it  is 
evident  they  must  still  increase  and  decrease  in  the  same 
manner  as  before. 

486.  If  then  a+b  be  raised  to  a  power  whose  exponent  is  n, 
The  exponents  of  a  will  be  ?i,  n-l,  n-2,  ...  .2,  1,   0  ; 

And  the  exponents  of  b  will  be  0,1,        2,  .  . .  .n-2,  n-l,   n. 

The  terms  in  which  a  power  is  expressed,  consist  of  the 
letters  with  their  exponents  and  the  co-efficients.  Setting 
aside  the  co-efficients  for  the  present,  we  can  determine,  from 
the  preceding  observations,  the  letters  and  exponents  of  any 
power  whatever. 

Thus  the  eighth  power  of  a+b,  when  written  without  the 
co-efficients,  is 

a8+a76+a6&3+a563+a464+a365+a266+a&7+&8. 


INVOLUTION      OP      BINOMIALS.  273 

And  the  nth  power  of  a+b  is, 

an+an"kb+an"aba a2bn~2  +a&n~1  +bn. 

Of  the  two  letters  in  a  term,  the  first  is  called  the  leading 
quantity,  and  the  other  the  following  quantity.  In  the  ex- 
amples which  have  been  given  in  this  section,  a  is  the  lead- 
ing quantity,  and  b  the  following  quantity. 

4L87.  The  number  of  terms  is  greater  by  1,  than  the  in- 
dex of  the  power.     For  if  the  index  of  the  power  is  n,  a  has, 
in  different  terms,  every  index  from  n  down  to  1 ;  and  there 
is  one  additional  term  which  contains  only  b.     Thus, 
The  square  has  3  terms,         The  4th  power,  5, 
The  cube  4,  The  5th  power,  6,  <fcc. 

488.  The  next  step  is  to  find  the  co-efficients.  This  part 
of  the  subject  is  more  complicated.  There  is,  however,  an 
easy  method  of  calculating  the  numerical  co-efficients  of  the 
powers  of  a  binomial,  to  some  extent  at  least,  by  actual  mul- 
tiplication. As  the  co-efficients  of  x-\-a  are  1  and  1,  if  these 
are  involved  by  themselves,  the  multiplier  is  continually  a 
unit.  In  arranging  the  terms  of  the  powers  of  a  binomial, 
according  to  the  exponents  of  the  two  letters,  (Art.  486,)  the 
particular  product,  from  multiplying  by  the  second  letter,  is 
carried  one  term  farther  to  the  right,  than  the  product  from 
multiplying  by  the  first.  If  this  arrangement  be  adopted,  in 
involving  1  +  1,  the  multiplication  will  be  effected  by  addition 
merely,  and  the  co-efficients  will  correspond  with  the  terms 
obtained  by  involving  the  letters. 

1  +  1 

1+   1 


The  square  1+2+   1  Whose  sum  is    4=2* 

1+  2+   1 


The  cube  1+3+  3+   1  8=23 

1+  3+  3+   1 


The  4th  power  1+4+  6+  4+   1  16=24 

1+  4+  6+  4  +  1 


The  5th  power  1  +  5+10+10+  5+1  32=2* 

1+  5+10  +  10+5+1 

The  6th  power  1+6+15+20+15+6  +  1  64=26 

* 


274  INVOLUTION      OF      BINOMIALS. 

In  the  same  manner,  if  the  co-efficients  of  the  terms  of 
any  power  of  a  binomial  be  given,  the  co-efficients  of  higher 
powers  may  be  easily  found. 

The  results  thus  obtained  are  the  co-efficients,  in  the  sev- 
eral terms  of  the  powers  of  x  +  a.  The  letters  are  easily 
arranged,  according  to  the  law  in  Arts.  485,  6.  If  then,  to 
the  several  terms,  we  prefix  their  corresponding  co-efficients, 
we  have  the  complete  power  of  the  binomial.     Thus 

(x+a)6=x6+Gx5a  +  l5x*a2+20x3a3  +  15x2a4-\-6xa5+a6. 

489.  By  recurring  to  the  numbers  in  Art.  488,  it  will  be 
seen,  that  the  co-efficients  first  increase,  and  then  decrease,  at 
the  same  rate ;  so  that  they  are  equal,  in  the  first  term,  and 
the  last,  in  the  second  and  last  but  one,  in  the  third  and  last 
but  two ;  and  universally,  in  any  two  terms  equally  distant 
from  the  extremes.  The  reason  of  this  is,  that  (a  +  b)n  is 
the  same  as  (b+a)n ;  and  if  the  order  of  the  terms  in  the 
binomial  root  be  changed,  the  whole  series  of  terms  in  the 
power  will  be  inverted. 

It  is  sufficient,  then,  to  find  the  co-efficients  of  half  the 
terms.     These  repeated  will  serve  for  the  whole. 

490.  In  any  power  of  (a+b),  the  sum  of  the  co-efficients 
is  equal  to  the  number  2  raised  to  that  power.  See  the  list 
of  co-efficients  in  Art.  488.  The  reason  of  this  is,  that,  ac- 
cording to  the  rules  of  multiplication,  when  any  quantity  is 
involved,  the  letters  are  multiplied  into  each  other,  and  the 
co-efficients  into  each  other.  Now  the  co-efficients  of  a ■+■& 
being  1  +  1=2,  if  these  be  involved,  a  series  of  the  powers 
of  2  will  be  produced. 

491.  To  determine  the  law  which  the  co-efficients  of 
any  term  of  a  binomial  uniformly  observe,  multiply  the  fac- 
tors x+a,  x+b,  x-\-c,  &c.  writing  the  several  co-efficients 
of  the  same  power  of  x  one  under  the  other. 

The  product  of  x-\-a  into  x+b  is 

4-«  ) 

i  >  x-\- ab.  This  into  x-\-c  gives 

fab  ) 

M-«c  >x+abc.     This  into  x+d  gives 
+  bc) 


INVOLUTION      OF      BINOMIALS. 


275 


+<0 


>x* 


+c 


-\-ac 
+  bc 

3 

-had 
+bd 
+  cdj 


+abc^ 
•\-abd 


>x2 


+acd  I 
+bcdj 


y  x-\-abcd. 


It  will  be  seen  that,  in  each  term  of  these  several  products, 
the  number  of  letters  in  one  combination  is  equal  to  the 
number  of  the  preceding  terms.     Also, 

In  the  first  term,  the  co-efficient  of  x  is  1. 

In  the  second,  it  is  the  sum  of  all  the  other  letters  of  the 
binomial  factors,  taken  singly. 

In  the  third,  it  is  the  sum  of  their  products,  taken  two  and  two. 
In  the  fourth,  it  is  the  sum  of  their  products,  taken  three 
and  three. 

The  last  term  is  the  product  of  all  these  letters. 

And  as  each  multiplication  introduces  a  new  letter  into  the 
several  combinations  which  constitute  the  co-efficients  of  x, 
the  relation  of  each  term  to  the  preceding  is  such,  that  if  m 
be  the  number  of  any  term,  and  r  be  the  number  of  the 
terms  which  precede  it,  the  co-efficient  of  x  in  this  mth  term, 
is  the  sum  of  the  several  combinations  of  the  other  letters  of 
the  binomial  factors  taken  r  and  r,  or  m—  1  and  m—  1. 
To  shew  this, 

492.  Let  A,  B,  G,  &c.  be  respectively  equal  to  the  dif- 
ferent co-efficients  of  x,  in  the  product  of  several  binomial 
factors.     This  product,  for  n  binomial  factors,  will  be 

xn+Axn~l+Bxn-2  +  Cxn-K..Nxn-r+'+Rxn-r...  +  U. 

As  in  the  successive  terms,  the  exponent  of  x  decreases 
by  1,  the  term  in  which  this  exponent  is  n  —  r  must  have  r 
terms  before  it ;  and  the  exponent  of  the  term  immediately 
preceding  must  be  n  —  r+l. 

Multiplying  the  given  product  by  an  additional  binomial 
factor  x-{-h,  we  have 


af*l+A)      +B    I    n  ,+C   )       o       R   ) 

+h  \      +Ah)x      +Bh)x      '"Nh)x 
which  is  the  product  of  7i-fl  binomial  factors 


n-r+i 


.+Uh, 


276  INVOLUTION      OF      BINOMIALS. 

As  multiplying  by  x  affects  not  the  co-efficients,  but,  in 
each  term,  adds  1  to  the  exponent ;  and  as  multiplying  by  h 
affects  not  the  exponents,  but  the  co-efficients  only ;  i\  fol- 
lows that,  in  arranging  the  new  product  of  n  +  1  binomials, 
according  to  the  powers  of  x,  we  derive  each  of  its  terms 
from  two  terms  of  the  previous  product  of  n  binomials,  now 
made  a  multiplicand.  Thus  the  co-efficient  of  the  third 
term  of  the  new  product  is  B-\-Ah  ;  that  is,  B  the  co-efficient 
of  the  third  term  of  the  multiplicand,  +  the  product  of  h 
into  the  co-efficient  of  the  second  term  of  the  multiplicand. 
As  B  is  the  sum  of  the  combinations  of  n  letters  taken  two 
and  two,  and  Ah  is  the  sum  of  the  same  letters  multiplied  by 
ft;  the  co-efficient  B+Ah  is  the  sum  of  the  combinations  of 
n-\- 1  letters,  taken  two  and  two. 

In  a  similar  manner,  the  term    jyi  I  xn~r+l    in  the  new 

product  of  n-\-l  binomials,  is  derived  from  the  two  terms 
Rxn~r  and  Nxn~7+l   in  the  previous  product  of  n  binomials. 

For       Rxn~r     X  x  =  Rx1l~r+  J 
And      Nxn~r+ "  X  h=Nhxn~r+  J 

As  R  is  the  sum  of  the  combinations  of  n  letters  taken  r  and  /*, 

And  Nh  is  the  sum  of  the  combinacions  of  these  letters 

taken  i — 1  and  r— 1,  and  multiplied  by  h  ;  the  co-efficient 

of  x  in  the  term    7%™  f  xn~r+  ]  is  the  sum  of  the  combinations 

Nh  )  J 

of  n+\  letters  taken  r  and  r.  % 

As  m  may  represent,  at  pleasure,  any  term  of  the  product 
of  any  number  of  binomial  factors,  the  above  law  of  the  co- 
efficients may  be  considered  universal. 

4:93.  To  make  the  application  to  the  powers  of  binomi- 
als, let  the  second  terms  of  the  factors  x+a,  x+b,  x+c,  &c. 
be  supposed  equal.  Then  will  ab=a2,  abc=a3,  abed—  a* , 
&c.     And  in  the  product  of  four  binomial  factors,  (Art.  491,) 


ab'} 

ac 

be 


>x*=4ax*. 


ad 
bd 
cd 


abc'} 
abd 
yx2=6a2x2.  >x=4a2x.    That  is, 

acd  I 
bed] 


INVOLUTION      OF      BINOMIALS.  277 

The  co-efficient  of  x*=a  repeated  as  many  times,  as  there 
are  letters  abed. 

The  co-efficient  of  x2=a2  repeated  as  often  as  there  are 
combinations  of  these  letters  taken  two  and  two. 

The  co-efficient  of  x=a3  repeated  as  often  as  there  are 
combinations  of  the  letters  taken  three  and  three. 

And  if  n  be  the  index  of  any  power  to  which  x+a  is  raised, 
The  co-efficient  of  xn~*=na. 

The  co-efficient  of  xn~2=a2  repeated  as  many  times,  as 
there  are  combinations  of  n  letters  taken  two  and  two ;  that 
.     n    7i—l 
is,  -X-^r-a2.     (Art.  483.) 

The  co-efficient  of  xn~3=a3  repeated  as  often,  as  there 
are  combinations  of  n  letters  taken  three  and  three ;  that  is, 
n    Ti—1     Ti—2 

Txirx-ira3- 

And  as  Rxll~r  is  the  term  which  has  r  terms  before  it, 
(Art.  491,)  the  co-efficient  of  xn~r  is  ar  repeated  as  often,  as 
there  are  combinations  of  n  letters  taken  r  and  r ;  that  is, 

_     P    n-r+1 

R=^X ar. 

Q,  r 

As  a,  in  this  formula,  is  the  following  quantity  of  the  bino- 
mial x+ a,  it  may  be  taken  separately  from  the  co-efficient 
p    n—r+\ 

of  x,  leaving  R=^X •     ' 

°  Q  r 

As  r  may  be  any  number  less  than  n,  this  law  may  be 
considered  as  applicable  to  any  term,  except  the  first,  of  the 
power  of  a  binomial. 

The  preceding  formula  may  be  resolved  into  three  factors, 

P  ,  1 

Tjj  7i— r-4-1,  and  -• 
Q,  r 

P  . 

As  jz  is  the  number  of  combinations  of  n  letters  taken 

r— 1  and  r— 1,  it  must  be  equal  to  the  co-efficient  of  the  pre- 
ceding term.     (Arts.  483,  491.) 

As  7i  — r  is  the  exponent  of  x  in  this  mth  term,  and  it 
decreases  regularly  by  1,  n— r-fl  must  be  equal  to  the  ex- 
ponent of  x  in  the  preceding  term. 

24 


278  INVOLUTION      OF      BINOMIALS. 

As  r  is  the  exponent  of  a  in  this  term,  and  it  increases 
regularly  by  1,  r—  1  must  be  the  exponent  of  a  in  the  pre- 
ceding term.  Therefore  the  co-efficient  of  any  term  is  equal 
to  the  co-efficient  of  the  preceding  term,  multiplied  ,  by  the 
exponent  of  the  leading  quantity  in  that  term,  and  divided 
by  the  exponent  of  the   following  quantity  increased  by  1. 

As  the  exponent  of  the  following  quantity  in  the  second 
term  is  1,  and  increases  regularly  by  1,  it  is  evident  that  in 
each  term,  it  is  equal  to  the  number  of  the  preceding  terms. 

If,  as  in  Art.  492,  (x+ a)n=xn+Axn-'  +Bxn~2 -\-Cxn~3,  &c. 

Then    A=n,  the  co-efficient  of  the  second  term, 

n—  1 
B=nX—^—>  of  the  third  term, 

<n i     ??  —  2 

C—nX— —  X-  ;~ »  of  the  fourth  term, 

71  ~~ —  1        7?      ■  2       71  —~  ^{ 

D=nX— —  X-x-X— —  >  of  the  tfftA  term,  &c. 
2  3  4  -^ 

By  the  method  in  Art.  488,  we  obtain  the  co-efficients  for 
the  several  terms  in  the  power  of  a  binomial,  from  the  next 
lower  power.  By  the  propositions  just  demonstrated,  we  ob- 
tain the  co-efficients  for  any  term  of  a  power  of  a  binomial 
from  the  preceding  term.  Connecting  these  with  the  law  of 
the  exponents,  as  given  in  Art.  486,  we  have  the  following 

Binomial  Theorem. 

491.  The  index  of  the  leading  quantity  of  the  power  of 
a  binomial,  begins  in  the  first  term  with  the  index  of  the  power, 
and  decreases  regularly  by  1.  The  index  of  the  following 
quantity  begins  with  1  in  the  second  term,  and  increases  regu- 
larly by  1. 

The  co-efficient  of  the  first  term  is  1  ;  that  of  the  second 
is  equal  to  the  index  of  the  power;  and  universally,  if  the 
co-efficient  of  any  term  be  multiplied  by  the  index  of  the  lead- 
ing quantity  in  that  term,  and  divided  by  the  iridex  of  the 
following  quantity  increased  by  1,  it  will  give  the  co-efficient 
of  the  succeeding  term. 

In  algebraic  characters,  the  theorem  is 

n  —  1 
(a+b)n=an+nXan-'b+nX  -^-an~2b2,  &c. 


INVOLUTION      OF      BINOMIALS.  279 

It  is  here  supposed,  that  the  terms  of  the  binomial  have  no 
other  co-efficients  or  exponents  than  1.  Other  binomials 
may  be  reduced  to  this  form  by  substitution. 

Ex.  1.  What  is  the  6th  power  of  x+y? 

The  terms  without  the  co-efficients,  are 
x6,  x5y,  x*y2,  x3y3,  x2y*,  xy5,  yQ. 
And  the  co-efficients,  are 

6X5     15X4     20X3 
h    *    ~2~~'    ~T~'    T"'    6'    L 
that  is,         1,    6,     15,         20,  15,        6,    1. 

Prefixing  these  to  the  several  terms,  wre  have  the  power 
required ; 

z6+6£5y+15o;42/2+20.z3y3  +  15x2y4+6^5+?/6. 

2.  (d+h)5  =d5  +5d*h+10d3  h2  +10d2  h3  +5dh*  +h5 . 

3.  What  is  the  nth  power  of  b-\~y? 

Ans.  bn+Abn-ly+Bbn-2y2-\- Cbn-3y3-\-Dbn-*y\&c. 
That  is,  supplying  the  co-efficients  which  are  here  repre- 
sented by  A,  B,  C,  &c. 

n x 

bn+nXbn'ly+nX-^- Xbn~2y2,  &c. 

4.  What  is  the  5th  power  of  x2  +3t/2  ? 
Substituting  a  for  x2,  and  b  for  3y2,  we  have 

(a+b)5  =a5  -\-5a*b  +  l0a3b2  +  10a2b3  +  5ab*  +b5 , 

And  restoring  the  values  of  a  and  b, 

(x2  +  Sy2)5  =x10  +  I5x9y2  +90x6yi  +210xiy6  +405x2y* 
+243?/ 10. 

5.  What  is  the  sixth  power  of  (Sx-\-2y)  ? 

Ans.  72to6+2910x'5?/+4860^4y2+43202r3?/3+2160x2?/4 
+576xy5+64y6. 

49 o.  A  residual  quantity  may  be  involved  in  the  same 
manner,  without  any  variation  except  in  the  signs.  By  re- 
peated multiplications,  as  in  Art.  230,  we  obtain  the  following 
powers  of  (a—b). 

(a-b)2=a2-2ab-hb2. 
(a-b)3=a3  —  Sa2b+Sab2—b3. 
(a-b)*=a*—4a3b+6a2b2-4ab3+b*,  <fcc. 


280  INVOLUTION      OF      BINOMIALS. 

By  comparing  these  with  the  like  powers  of  (a-\-b)  in  Art. 

485,  it  will  be  seen,  that  there  is  no  difference  except  in  the 

signs.     There,  all  the  terms  are  positive.     Here,  the  terms 

which  contain  the  odd  powers  of  b  are  negative.   See  Art.  233. 

The  sixth  power  of  (x—y)  is 

x6  —  Gx5y+15xAy2  —  20x3y3  +  15x2yA—6xy5+y6. 
The  ?Uh  power  of  (a—b)  is 

an—Aan-1b+Ban-2b2  —  Can'3b3t  &c. 

496.  When  one  of  the  terms  of  a  binomial  is  a  unit,  it 
is  generally  omitted  in  the  power,  except  in  the  first  or  last 
term ;  because  every  power  of  1  is  1,  (Art.  226,)  and  this 
when  it  is  a  factor,  has  no  effect  upon  the  quantity  with 
which  it  is  connected.     (Art.  85.) 

Thus  the  cube  of  (x+1)  is         x3+3x2  Xl+3.rXl2+l3} 
Which  is  the  same  as  x3-\-3x2 +3x+l. 

The  insertion  of  the  powers  of  1  is  of  no  use,  unless  it  be 
to  preserve  the  exponents  of  both  the  leading  and  the  follow- 
ing quantity  in  each  term,  for  the  purpose  of  finding  the  co- 
efficients. But  this  will  be  unnecessary,  if  we  bear  in  mind, 
that  the  sum  of  the  two  exponents,  in  each  term,  is  equal  to 
the  index  of  the  power.  (Art.  485.)  So  that,  if  we  have  the 
exponent  of  the  leading  quantity,  we  may  know  that  of  the 
following  quantity,  and  v.  v. 

Ex.  1.  The  sixth  power  of  (1—  y)  is 

1  —  Gy+15y2  —  20y3  +  15y*—Gy5 +y*. 
2.     (l+x)n=l-hAx+Bx2  +  Cx3+Dx\  &c. 

497.  From  the  comparatively  simple  manner  in  which 
the  power  is  expressed,  when  the  first  term  of  the  root  is  a 
unit,  is  suggested  the  expediency  of  reducing  other  binomials 
to  this  form. 

The  quotient  of  (a+x)  divided  by  a  is  (n--)*  This  mul- 
tiplied into  the  divisor,  is  equal  to  the  dividend;  that  is, 
(a+x)=axyl+-y,   therefore  (a  +  x  )n  =  an  x(l+-)  - 

By  expanding  the  factor  (l  +  ~)  >  we  have 

(x\n  /         x        x2        x3  \ 

1-f-)  =anX  [1+A-  +  &—  +  C— .)  &c. 
af  \         a        a2        a3  / 


INVOLUTION      OF      BINOMIALS.  281 

498.  LetA+Bx+Cx2+Dx*+&Lc.=M+Nx+Px*+Qx*+ 
&c.  or  (Art.  183,) 

(A-M)  +  (B-N)x+(C-P)x*  +  (D-Q)x*+  &c.  =0. 

An  equation  of  this  form  is  said  to  be  identical,  if  the 
equality  subsists,  whatever  value  be  given  to  x.  (Art.  173.) 
In  such  an  equation,  the  co-efficients  of  the  several  powers 
of  x,  as  well  as  the  aggregate  of  the  terms  into  which  x  does 
not  enter,  must  each  =0.     That  is, 

A-M=Q,  B-N=Q,  C-P=0,  &c. 

For  if  the  equation  holds  true,  whatever  be  the  value  of  x, 
it  must  do  so,  when  x  is  an  infinitesimal.  But  on  this 
supposition, 

(B-N)x  +  (C-P)x2  ±(D-Q)x*+  &c.  each=0.  (Art. 469.) 
Unless,  then,  in  the  equation, 
(A-M)  +  (B-N)x+(C-P)z*  +  (D-Q)x*  +  &c.  =0, 

the  remaining  term  (A  —  M)  —  0,  the  whole  of  this  member 
of  the  equation  can  not  =0. 

Dividing    (B-N)x+(C-P)x2  +  (D-Q)x*+  &c.  bv  x, 
We  have  (B-N)  +(C-P)x+(D-Q)x2  +  &c.  =0. 
Therefore  (B-N)=0.  Dividing  again  by  a:,  we  have  (C-P)=0. 
Hence  A=M,  B=N,  C=P,  D=Q,  &c. 

4:90.  In  the  demonstration  which  has  been  given  of  the 
binomial  theorem  (Arts.  491,  2,  3,)  the  exponent  of  the  bino- 
mial has  been  supposed  to  be  a  positive  whole  number.  But 
the  theorem  is  applicable  to  cases  in  which  the  exponent  is 
fractional,  and  either  positive  or  negative.     To  prove  this, 

m 

Let     (l+x)n  =  l+Ax+Bx2-\-Cx*+  &c. 

M 

And    (l+y)"  =  l+Ay  +  By*  +  Cy*  +  &c. 
A,  B,  C,  &c.  being  the  co-efficients  whose  values  are  to 
be  determined. 

Subtracting  one  of  these  series  from  the  other,  we  have 

m  m. 

{\+x)^-(\+yy=A.{x-y)  +  B.{x2-ij2)  +  C.(x*-y:i)+&ic. 
Put  (1+2?)==**,  and  (l+y)=v\     Then  zn—vn=--x-y. 
Applying  the  exponent  ?  to  both  members  of  the  equations 
(l+x)  =  zn,  and  (l+y)  =  vn,  (Art.  234,)  we  have 

m  m 

(l+x)"=zm,  and  (l+y)n=vm.     Therefore, 
zm~v™=A.  (x-y)+B .  (x2-y2)  +  C .  (x*-y3)+  &c. 

24* 


282  INVOLUTION      OP      BINOMIALS. 

And  as  zn—vn=x—y, 

y**___  m|M  <>»2_.<»y2  'p3___^#3 

=^+2?. — +  C. —  +  &c.  which  is  equal  to 

zn— vn  %—y  x~-y 

A+B.(x+y)  +  C.(x2+xy+y2)  +  &c.    (Art.  130.) 
Dividing  the  numerator  and  denominator  of  the  fraction 

— by  z—v,  (Art.  130,)  we  have 

ytn- 1  _i_  y m-2  7«  _i_  y in-  3  *%  2      nstn- 1 

z — -rz     v-rz      c      v        A+B  /   .    )    C(  >+      .  y2)+&c 

As  a;  and  y  may  be  of  any  value,  let  #=y. 

Then  z=u,  each  term  in  the  numerator  becomes  z™""1,  and 

each  in  the  denominator   zn_1.     Observing  then,  that  the 

number  of  terms  in  one  =m}  and  in  the  other  =n,  (Art.  130,) 

we  have 

mzm-1      mzm  n         _ 

— —  =— -=A+2Bx+3Cx2+&,c. 

nz11   1        nzn 

Multiplying  the  first  member  of  the  equation  by  z",  and  the 

m 

second  by  l-\-x  which  —zn\  and  observing  that  zn=[l+x)ns 

m  - 

we  have       ~-.(l+x)n=A+2Bx+3Cx2  +  <fcc.  ) 

+  Ax+2Bx2  +  &c.  ) 

771 

But  multiplying  by  —  the  equation  at  the  beginning  of  this 

article,  we  have 

-,(l+x)n  =  -  +  -Av+-Bx2+-Cx3  +  <fcc. 

As  the  members  of  these  two  equations  are  equal,  (Ax.  11,) 
and  as  the  powers  of  x,  in  the  corresponding  terms  are  the 
same  in  both,  the  co-efficients,  which  are  independent  of  any 
particular  values  of  x,  must  be  equal,  the  first  term  of  one 
feeing  equal  to  the  first  term  of  the  other,  the  second  of  one, 
to  the  second  of  the  other,  &c.  (Art.  498.)  By  comparing 
them,  we  have 

—  =A  Hence  A=— 

n  n 

m  m 

*A=*B+A  B=A.(^)M^-) 

n  2  7i      2 


INVOLUTION      OF      BINOMIALS.  283 

m  mm 

™B=3C+2B  e-i/.(vWV  (V) 

n  3  n      2  3 

This  result  agrees  with  the  rule  for  the  co-efficients,  in  the 

binomial  theorem.     (Art.  494.) 

m 
If  n  be  put  =1,  the  exponent  —=m}  a  positive  whole 

number.  m 

And  A=m,  the  co-efficient  of  the  second  term. 

B=ml— —  f  of  the  third  term. 

C=m(  — - — J  ( — —  )  of  the  fourth  term,  as  in  Art.  493. 

•500.  The  demonstration  is  nearly  the  same,  when  the 

m 
exponent  of  the  binomial  l+x  is  negative, being  substi- 

m 
tuted  for  — ,  and  due  regard  being  paid,  at  every  step,  to  the 

sign  of  the  exponent. 

Let     (l+x)    n=l+Ax+Bx2  +  Cx*  +  &c. 

__m 

And    (1+y)    n  =  1+Ay+By*  +  Cy*  +  &c. 
By  subtraction, 

(l+x)"^(l+t/rn=^.(^-y)+^.(^3-y2  +  C7).(x3--2/3)  +  &c. 
Put  (l+#):=zn,  and  (l+y)=vn.     Then  zn—vn=x—y. 

mm. 

Applying  the  exponent 1  as  H —  was,  in  Art.  499,  we  have 

(l—x)    *=*-*,  and   (1+y)    n  =v~m.     Therefore, 

z-m-v-m=A.(x-y)+B.(x2—y2)  +  C.(x3—y3)+  &c. 

And  as  zn— v1l=x— y, 

n__  n    =A+B.(x+y)  +  C.(x2  +xy+y2)  +  &c.  (Art.  J 30.) 

1      1  um-zw 

The  numerator  2r«_tr«=— -—  (Art. 224)  =  _-(ArU50.) 

Hence  — —  =  ^%-^^j=-^ir:^j,(Artl48.) 


284  INVOLUTION      OF      BINOMIALS. 

It  has  been  shown,  (Art.  499,)  that  when  z=v, 
zm_^vm  mzm~l  mzm 

becomes  — — ■ ->  or 


zn  —  vn  nzn~l  nzn 

On  the  same  supposition, 

z~m— v~m  _        1  /mzm-l\         mz-m-l_      mz'm 
~^-vw        ~"  zam\nzn"1/  nz""1  nzn 

Therefore, 

zn_vn     or  —^=A+B.(x+y)  +  C.(z'+zy+y*)  +  &c. 

Multiplying  the  first  member  of  the  equation  by  znf  and  the 
second  byl  +x  which  =zn,  and  observing  that  z~m=(l+x)    * 
we  have 


(1+x)    n  =A+2Bx+SCx2  +  &c 

+  Ax+2Bx2  +  &c 


i 


TJX 

Multiplying  by the  equation  at  the  beginning  of  this 

article, 

m     /,  v"?  m        m  i  mn  m^ 

(1+x)     = Ax Bx* Cz3-&c. 

n    v        '  n       n  n  n 


Comparing  the  two  equations,  we  have 


=A    Therefore  A= 

n  n 

m  m 

n  2  7i        2 

m  m  m 

-™B=3C+2B         C=b{~^)=H^){~-^) 

n  o  71         2>  S 

The  co-efficient  of  the  second  term  is  negative. 

That  of  the  third  term,  being  the  product  of  two  negative 

factors  is  positive. 

That  of  the  fourth  term,  the  product  of  three  negatives  is 

negative.     That  is.  when  a  binomial  whose  exponent  is  a 

negative  fraction  is  expanded,  the  co-efficients  of  the  several 

terms  are  alternately  positive  and  negative. 


INVOLUTION      OF      BINOMIALS.  285 

m 

501.  The  exponent >  when  n=l,  becomes  —m,  a 

negative  whole  number ;  and 

A=  —  m,    the  co-efficient  of  the  second  term. 

B=—  ml — - — j  of  the  third  term. 

C=  —  mi — - — )( — - — )  of  the  fourth  term. 

The  same  as  in  Art.  493,  except  that  the  co-efficients  here 
are  alternately  positive  and  negative. 

We  have  then,  in  this  and  preceding  articles,  demonstra- 
tions of  the  binomial  theorem,  for  each  of  the  four  cases,  in 
which  the  exponent  of  the  binomial  is  either  a  positive  whole 
number,  a  positive  fraction,  a  negative  fraction,  or  a  nega- 
tive whole  number. 

502.  When  the  index  of  the  power  to  which  any  bino- 
mial is  to  be  raised  is  a  positive  whole  number,  the.  series  will 
terminate.  The  number  of  terms  will  be  limited,  as  in  all 
the  preceding  examples. 

For,  as  the  index  of  the  leading  quantity  continually  de- 
creases by  one,  it  must,  in  the  end,  become  0,  and  then  the 
series  will  break  off. 

Thus  the  5th  term  of  the  fourth  power  of  a+x  is  xx,  or 
«°:c4,  a0  being  commonly  omitted,  because  it  is  equal  to  1. 
(Art.  224.)  If  we  attempt  to  continue  the  series  farther,  the 
co-efficient  of  the  next  term,  according  to  the  rule,  will  be 

1 XO 

— —  =  0.     (Art.  106.)     And  as  the  co-efficients  of  all  suc- 
ceeding terms  must  depend  on  this,  they  will  also  be  0. 

503.  If  the  index  of  the  proposed  power  is  negative,  this 
can  never  become  0,,by  the  successive  subtractions  of  a  unit. 
The  series  will,  therefore,  never  terminate;  but  like  many  deci 
mal  fractions,  may  be  continued  to  any  extent  that  is  desired. 

Ex.    Expand  into  a  series   j-~r- T^  —  {aJry)~2- 

The  terms  without  the  co-efficients,  are 

a"2,  a~3y,  a~Ay2,  a'5y3f  a~6y4,  &c. 

The  co-efficient  of  the  2d  term  is  -2,  of  the  4th - =  — 4. 


286  INVOLUTION      OP      BINOMIALS. 

Of  the  third,  Z^~L  =  +3,  of  the  5th,  Z*2Lzl  =  +5. 

The  series  then  is 

a"2—  2tfT32/-f3tfr4z/2—  4dT5?/3+5dr6?/4,  &c. 

Here  the  law  of  the  progression  is  apparent ;  the  co-effi- 
cients increase  regularly  by  1,  and  their  signs  are  alternately 
positive  and  negative. 

tfO-1.  The  Binomial  Theorem  is  of  great  utility,  not  only 
in  raising  powers,  but  particularly  in  finding  the  roots  of  bino- 
mials. A  root  may  be  expressed  in  the  same  manner  as  a 
power,  except  that  the  exponent  is,  in  the  one  case  an  integer, 
in  the  other  a  fraction.  (Art.  256.)  Thus  (a+b)n  may  be  either 
a  power  or  a  root.     It  is  a  power  if  n=2,  but  a  root  if  n=\. 

♦">0o.  If  a  root  be  expanded  by  the  binomial  theorem,  the 
series  will  never  terminate.  A  series  produced  in  this  way 
terminates,  only  when  the  index  of  the  leading  quantity  be- 
comes equal  to  0,  so  as  to  destroy  the  co-efficients  of  the  suc- 
ceeding terms.  (Art.  502.)  But  according  to  the  theorem, 
the  difference  in  the  index,  between  one  term  and  the  next, 
is  always  a  unit ;  and  a  fraction,  though  it  may  change  from 
positive  to  negative,  can  not  become  exactly  equal  to  0,  by 
successive  subtractions  of  units.  Thus,  if  the  index  in  the 
first  term  be  -|,  it  will  be, 

In  the  2d,      i-l  =  -£,         In  the  4th,  —  f  —  1  =  —  f, 
In  the  3d,  —  \— 1  =  — f,         In  the  5th,  —  f-l  =  -J,  &c. 
Ex.   What  is  the  square  root  of  (a-{-b)  ? 
The  terms,  without  the  co-efficients,  are, 

1  —1  —1  __5  _7 

a2,  a   2b,  a   2b2,  a   2b3,  a   2b*,  &c. 
The  co-efficient  of  the  second  term  is     4-A, 


D 


*~~i_      i    „e  *u-.  A*u    ~JX~~Ji 


of  the  3d,  a-y-£=-|,  of  the  4th,  -  --,-„. 

And  the  series  is  a^-\-\cT^b— ^a"2b2  +  T\a'~h3>  &c. 

When  a  quantity  is  expanded  by  the  Binomial  Theorem,  the 
law  of  the  series  will  frequently  be  more  apparent,  if  the  fac- 
tors, by  which  the  co-efficients  are  formed,  are  kept  distinct. 

1.  Expand  into  a  series   (a2+z)^. 


INVOLUTION      OF      BINOMIALS.  287 

Substituting  b  for  a2,  we  have 
(b-\-x)^=b^-hAb"K+Bb"^x2  +  Cb^x3+Db^xAf  &c. 

^~2X   2    ~2X      4~      2.4" 
C--—     Zl-      JL      -?  3 


2.4       3  2.4  6       2.4.6 

^         3         -#  3  5  3.5 


2.4.6       4         2.4.6         8  2.4.6.8 

Restoring,  then,  the  value  of  b}  and  writing  -  for  a"1,  we  have 


a 


i  a;  x2  Sx3  3.5a:4 

(a,+x)*=a+____+______,  &c. 

i 

2.  Expand  into  a  series   (l+z)^. 

x       x2         Sx3  3.5a?4 

AnS-    1  +  2""^4  +  2X6-2X6T8J&C- 

3.  Expand    </2,  or  (1  +  1)*. 

1      J_  3  3.5  3.5.7 

AnS'  1  +  2      2.4  +  2.4.6      2.4.6.8  +  2.4.6.8.10'&C' 

4.  Expand  (a+xfi,  or/x(l+Jl     See  Art.  497. 

i     /         x  x2  Sx3  3.5a:4         .     \ 

\       2a      2.4a2      2.4.6a3       2.4.6.8a4  / 

5.  Expand  (a+bfc,  or  a?X\l+--jk 

i    (      A        2ft2  2.5&3  2.5.8ft4  \ 

Ans.  a  XV1^-^^*  3.6.9^  ~OXT^'&C7 

i 

6.  Expand  into  a  series   (a— &)?. 

J    /        6         Sb2  3.7b3  3.7. 11&4  \ 

Ans.  a   X^l     ^      4^2       4^12a3      4.8.12.16a4'&C7 

1  2 

7.  Expand  (a+a:)  »      8.  Expand  (1—  a:)5. 
9.  Expand  (l+a:)~i     10.  Expand  (a2+a:)~* 


288  INVOLUTION      OF    BINOMIALS. 

•"500.  The  binomial  theorem  may  also  be  applied  to  quan- 
tities consisting  of  more  than  two  terms.     By  substitution, 
.  several  terms  may  be  reduced  to  two,  and  when  the  com- 
pound expressions  are  restored,  such  of  them  as  have  expo- 
nents may  be  separately  expanded. 

Ex.    What  is  the  cube  of  a+b+c? 
Substituting  h  for  (b+c),  we  have  a+(b+c)=a+h. 
And  by  the  theorem,  (a+h)3=a3+3a2h+3ah2+h3. 
That  is,  restoring  the  value  of  h, 
(a+b+c)3=a3+3a2X(b+c)+3aX(b+c)2+(b+c)3. 

The  two  last  terms  contain  powers  of  (b+c)  \  but  these 
may  be  separately  involved. 

Promiscuous  Examples. 

1.  What  is  the  8th  power  of  (a+b)  ? 

Ans.  a8+8a7&+28a6&2+56a5&3+70a4&4+56a3&5-f 
28a2b°+8ab1+b3. 

2.  What  is  the  7th  power  of  (a—b)? 

3.  Expand  into  a  series    — —  >  or  (1— a)"1. 

Ans.  l+a+a24-a3+a4+a5,  &c. 

4.  Expand r>  or  hx(a— ft)"1. 

7     /l      b     b2     b3         \        h    bh    b2h    b3h    , 
Ans.  Ax(-+— +~  +—  >&c.)  or-+— +— +— >&c 
\a    a2     a3     a*  la     a2      a3       a4 

5.  Expand  into  a  series   (a2+b2)^. 

b2      b*        b«      . 

Ans.  a+- ^-t+ttt-t'  &c. 

2a     8a3     16a5 

6.  Expand  into  a  series  (a+y)"4. 


1      4?/     10y2     20y3     35y4     . 
a4     a5       a6         a7         a8 


7.  Expand  into  a  series   (c3-frr3)* 

/       x3       2x6        2.5x9      0     \ 
An,  cX(l+--— +3— -,  &c.) 


EVOLUTION     OF     COMPOUND     QUANTITIES.    289 
d 


a  l 

8.  Expand     ,     .  or  d(c2+z2)*. 

r  Vc2+x2  v  ' 

dl        x2         SxA  3.5x8  3.5.7x8  \ 

AnS*   cl1~^  +  2^^^2X6^  +  2.4.6.8c«'&C-; 

9.  Find  the  5th  power  of  (a2+y3). 

10.  Find  the  4th  power  of  (a+b+x). 

11.  Expand   («3-x)^.  12.  Expand   (1-y2)^. 


13.  Expand   (a-^^f^^^F^^xpand  A(a3-y3)f 


SECTION    XVII. 

EVOLUTION"    OF    COMPOUND    QUANTITIES. 

Art.  507.  The  roots  of  compound  quantities  may  be 
extracted  by  the  following  general  rule : 

After  arranging  the  terms  according  to  the  powers  of  one 
of  the  letters,  so  that  the  highest  power  shall  stand  first,  the 
next  highest  next,  &c. 

Take  the  root  of  the  first  term,  for  the  first  term  of  the 
required  root : 

Subtract  the  power  from  the  given  quantity,  and  divide  the 
first  term  of  the  remainder,  by  the  first  term  of  the  root  in- 
volved  to  the  next  inferior  power,  and  multiplied  by  the  index 
of  the  given  power.;*  the  quotient  will  be  the  next  term  of  the 
root. 

Subtract  the  power  of  the  terms  already  found  from  the 
given  quantity,  and  using  the  same  divisor,  proceed  as  before. 

This  rule  verifies  itself.  For  the  root,  whenever  a  new 
term  is  added  to  it,  is  involved,  for  the  purpose  of  subtracting 

*  By  the  given  power  is  meant  a  power  of  the  same  name  with  the  required 
root.  As  powers  and  roots  are  correlative,  any  quantity  is  the  6quare  of  its 
square  root,  the  cube  of  its  cube  root,  <fec. 

25 


290       EVOLUTION     OF     COMPOUND     QUANTITY 

its  power  from  the  given  quantity :   and  when  the  power  is 
equal  to  this  quantity,  it  is  evident  the  true  root  is  found. 

Ex.  1.   Extract  the  cube  root  of 

a6+3a5--3a4--lla3+6a2  +  12a-8(a2-f-a--2. 
a6,  the  first  subtrahend. 


3a 4)*     3a5,     &c  the  first  remainder. 

a 6  +  3a 5  -f  3a 4  -f  a 3 ,    the  second  subtrahend. 


3a4)#     *       —6a4,     &c.  the  second  remainder. 

a6+3a5-3a4-lla3+6a2  +  12a-8. 

Here  a2,  the  cube  root  of  a6,  is  taken  for  the  first  term  of 
the  required  root.  The  power  a8  is  subtracted  from  the 
given  quantity.  For  a  divisor,  the  first  term  of  the  root  is 
squared,  that  is,  raised  to  the  next  inferior  power,  and  multi- 
plied by  3,  the  index  of  the  given  power. 

By  this,  the  first  term  of  the  remainder  3a5,  &c.  is  divided, 
and  the  quotient  a  is  added  to  the  root.  Then  a2  -fa,  the 
part  of  the  root  now  found,  is  involved  to  the  cube,  for  the 
second  subtrahend,  which  is  subtracted  from  the  whole  of  the 
given  quantity-  The  first  term  of  the  remainder  —6a*,  &c. 
is  divided  by  the  divisor  used  above,  and  the  quotient  —2  is 
added  to  the  root.  Lastly  the  whole  root  is  involved  to  the 
cube,  and  the  power  is  found  to  be  exactly  equal  to  the  given 
quantity. 

It  is  not  necessary  to  write  the  remainder  at  length,  as,  in 
dividing,  the  first  term  only  is  wanted. 

2.  Extract  the  fourth  root  of 

a4+8a34-24a2+32a+16(a+2 


4a3)*     8a3,   &c. 


a4+8a3+24a2+32a+16. 

3.  What  is  the  fifth  root  of 

as+5a4/?+10a3&2  +  10a3&3+5a&4+&5?     Ans.  a+b. 

4.  What  is  the  cube  root  of 

a3-Ga26+12aJ2-8&3?  Ans.  a-26. 


EVOLUTION     OF     COMPOUND     OUANTITIES.    291 

5.  What  is  the  square  root  of 

4a2  -12ab+9b2  +  16ah-24bh+Wt2  (2a-3b+4h 
4a2 


4a)*-12ab,  &c. 
4a2-12ab+9b2 


4a)*       *       *+        I6ah,  &c. 


4a2  —  12ab+9b2  +  \Gah-24bh  +  16h2. 

In  finding  the  divisor  here,  the  term  2a  in  the  root  is  not 
involved,  because  the  power  next  below  the  square  is  the 
first  power. 

508.  But  the  square  root  is  more  commonly  extracted 
by  the  following  rule,  which  is  of  the  same  nature  as  that 
which  is  used  in  Arithmetic. 

After  arranging  the  terms  according  to  the  powers  of  one 
of  the  letters,  take  the  root  of  the  first  term,  for  the  first  term 
of  the  required  root,  and  subtract  the  power  from  the  given 
quantity. 

Bring  down  two  other  terms  for  a  dividend.  Divide  by 
double  the  root  already  found,  and  add  the  quotient,  both  to 
the  root,  and  to  the  divisor.  Multiply  the  divisor  thus  in- 
creased, into  the  term  last  placed  in  the  root,  and  subtract 
the  product  from  the  dividend. 

Bring  down  two  or  three  additional  terms  and  proceed  as 
before. 

Ex.  1.  What  is  the  square  root  of 

a2  +2ab+b2  +2ac+2bc+c2  (a+b+c. 
a2,  the  first  subtrahend. 

2a+b)*      2ab+b2 

Into  b==      2ab-\-b2,  the  second  subtrahend. 


2a+2b+c)   *     *     2ac+2bc+c2 

Into  c=  2ac+2bc+c2i  the  third  subtrahend. 

Here  it  will  be  seen,  that  the  several  subtrahends  are  suc- 
cessively taken  from  the  given  quantity,  till  it  is  exhausted.  If 
then,  these  subtrahends  are  together  equal  to  the  square  of  the 
terms  placed  in  the  root,  the  root  is  truly  assigned  by  the  rule. 

The  first  subtrahend  is  the  square  of  the  first  term  of  the  root. 


292      EVOLUTION     OF     COMPOUND     QUANTITIES. 

The  second  subtrahend  is  the  product  of  the  second  term 
of  the  root,  into  itself,  and  into  twice  the  preceding  term. 

The  third  subtrahend  is  the  product  of  the  third  term  of 
the  root,  into  itself,  and  into  twice  the  sum  of  the  two  pre- 
ceding terms,  &c. 

That  is,  the  subtrahends  are  equal  to 

a2  +  (2a+b)xb+(2a+2b+c)Xc,  kc. 
and  this  expression  is  equal  to  the  square  of  the  root. 

For  (a+b)2=a2+2ab+b2=a2  +  (2a+b)Xb.     (Art.  119.) 

And  putting  h=a+b,  the  square  h2=a2 +  (2a+b)xb. 

And   (a+b+c)2  =  (h+c)2=h2  +  (2h+c)Xc; 
that  is,  restoring  the  values  of  h  and  h2, 

(a+b+c)2  =  a2  +  (2a+b)xb+(2a+2b+c)Xc. 

In  the  same  manner,  it  may  be  proved,  that,  if  another 
term  be  added  to  the  root,  the  power  will  be  increased,  by 
the  product  of  that  term  into  itself,  and  into  twice  the  sum 
of  the  preceding  terms. 

The  demonstration  will  be  substantially  the  same,  if  some 
of  the  terms  be  negative. 

2.  What  is  the  square  root  of 

1  -4b+4b2  +2y-4by+y2  (1  -2b+y 
1 

2-2b)*-4b+ib2 

Into     —  2b=—  4b+4b2 

i 

2— 4b+y)  *      *      2y—4by+y2 
Into  y—  2y—4by+y2 

3.  What  is  the  square  root  of 

a6-2a5+3a<-2a3+a2  ?  Ans.   a3-a2+a. 

4.  What  is  the  square  root  of 

a*  +4a2b+4b2  -4a2  -8b+4?        Ans.  a2 +2b-2. 

It  will  frequently  facilitate  the  extraction  of  roots,  to  con- 
sider the  index  as  composed  of  two  or  more  factors. 

Thus  a*=a%Xk  (Art. 269.)  And  a^=JX^.  That  is, 
The  fourth  root  is  equal  to  the  square  root  of  the  square  root; 
The  sixth  root  is  equal  to  the  square  root  of  the  cube  root; 
The  eighth  root  is  equal  to  the  square  root  of  the  fourth 
root,  &c. 


EVOLUTION      OF     COMPOUND      ftUANTITIES.     293 

To  find  the  sixth  root,  therefore,  we  may  first  extract  the 
cube  root,  and  then  the  square  root  of  this. 

1.  Find  the  square  root  of  x4— 4x3+6a;2—  4#+l. 

2.  Find  the  cube  root  of  x*-6x5  +  15x*-20xz-{-15x2-Gx+l. 

3.  Find  the  square  root  of  4x*  —  4x3  +  13x2  —  Gx  +  9. 

4.  Find  the  4th  root  of  M>a*-96a*x+216a2x2-216ax3-t-81x4 . 

5.  Find  the  5th  root  of  x5 +5x*  +  10x*  +  10x2 +5x+l. 

6.  Find  the  6th  root  of 

a6-6a5b+l5a*b2-20a3b3  +  15a2b*-6ab*+b*. 

Roots  of  Binomial  Surds. 

509.  It  is  sometimes  expedient  to  express  the  square  root 
of  a  quantity  of  the  form  adtz  y/b,  called  a  binomial  or  re- 
sidual surd,  by  the  sum  or  difference  of  two  other  surds.  A 
formula  for  this  purpose  may  be  derived  from  the  following 
propositions : 

1.  The  square  root  of  a  whole  number  can  not  consist  of 
two  parts,  one  of  which  is  rational,  and  the  other  a  surd. 

If  it  be  possible,  let   y/a=x+  y/y,  in  which  the  part  x  ia 

rational. 

Squaring  both  sides,   a=x2  +2xy/y+y 

a — x2  —  t/ 
And  reducing,  Vy= « >  a  rational  quantity; 

which  is  contrary  to  the  supposition. 

2.  In  every  equation  of  the  form  x-\-  y/y=a-\-  y/b,  the 
rational  parts  on  each  side  are  equal,  and  also  the  remaining 
parts. 

If  x  be  not  equal  to  <z,  let  x=ad=:z. 
Then  a±:z+ y/y=a+ y/b.  And   y/b=z+y/y, 

That  is,  y/b  consists  of  two  parts,  one  of  which  is  rational, 
and  the  other  not ;  which,  according  to  the  preceding  propo- 
sition, is  impossible. 

In  the  same  manner  it  may  be  shewn,  that  in  the  equation, 
x—  y/y— a—  y/b,  the  rational  parts  on  each  side  are  equal, 
and  also  the  remaining  parts. 

3.  If  \/a+y/b=x+  y/y,         then   Va-  y/b=x-  y/y. 
For,  by  squaring  the  first  equation,  we  have 

a+y/b=x2+2xy/y+y 

25* 


294    EVOLUTION     OP     COMPOUND     aUANTITIES 

And  by  the  last  proposition, 

a=x2  +y 
y/b—2x^y 


a 


By  subtraction,  a—  ^/b—x2  —2x >/y+y 

By  evolution,  </a—  i/b=x—  x/y 

510.  To  find,  now,  an  expression  for  the  square  root  of 
binomial  or  residual  surd, 

Let  Va+^b=x+y/y 

Then  *J  a—  y/b=x—  y/y 

Squaring  both  sides  of  each,  we  have 

a+  y/b=x2  +2x^/y+y 

a—  x/b—x2—2xy/y+y 

Adding  the  two  last,  and  dividing,  a=x2  -\-y 

Multiplying  the  two  first,  y/a2—b=x2—y 

Adding  and  subtracting, 

a+V^Tb=2x*.     Or  z=\/?±^*E± 
a-V^=b=2y     And  Vy=\/-— ^H* 


Therefore,  as  v,«+N/6=x+v/y,  and  *J a— s/b=x— y/y, 


2  V  2 

Or,  substituting  d  for  V 'a2  —b, 


1.  Va+y/b=^^(a+d)+x/l(a-d). 

2.  \/<z-  ^b=  V\(a+d)-y/\(a-d). 

Ex.  1.   Find  the  square  root  of  3+2v/2. 

Here  a=S,  a2=9,  ^6=2^2,  6=8,  a2-b=9-8=l. 

Therefore   ^3+2^2=  V/?±l  +  N/^Zl==  v2  +  l. 

2.  Find  the  square  root  of  ll+6v/2.  Ans.  3+^2. 


INFINITE      SERIES.  295 

3.  Find  the  square  root  of  6—2  v/5.  Ans.   v/5—  1. 

4.  Find  the  square  root  of  7+4  v/3.  Ans.  2+^/3. 

5.  Find  the  square  root  of  7— 2^/10.       Ans.   v/5—  y/2. 

These  results  may  be  verified,  in  each  instance,  by  multi- 
plying the  root  into  itself,  and  thus  re-producing  the  binomial 
from  which  it  is  derived. 


SECTION    XVIII. 

INFINITE    SERIES. 


Art.  S\\*  It  is  frequently  the  case,  that,  in  attempting 
to  extract  the  root  of  a  quantity,  or  to  divide  one  quantity 
by  another,  we  find  it  impossible  to  assign  the  quotient  or 
root  with  exactness.  But,  by  continuing  the  operation,  one 
term  after  another  may  be  added,  so  as  to  bring  the  result 
nearer  and  nearer  to  the  value  required.  When  the  number 
of  terms  is  supposed  to  be  extended  beyond  any  determinate 
limits  the  expression  is  called  an  infinite  series.  The  quan- 
tity, however,  may  be  finite,  though  the  number  of  terms  be 
unlimited. 

An  infinite  series  may  appear,  at  first  view,  much  less  sim- 
ple than  the  expression  from  which  it  is  derived.  But  the 
former  is,  frequently,  more  within  the  power  of  calculation 
than  the  latter.  Much  of  the  labor  and  ingenuity  of  mathe- 
maticians has,  accordingly,  been  employed  on  the  subject  of 
series.  If  it  were  necessary  to  find  each  of  the  terms  by 
actual  calculation,  the  undertaking  would  be  hopeless.  But 
a  few  of  the  leading  terms  will,  generally,  be  sufficient  to 
determine  the  law  of  the  progression. 

•5155.  A  fraction  may  often  be  expanded  into  an  infinite 
series,  by  dividing  the  numerator  by  the  denominator.  For 
the  value  of  a  fraction  is  equal  to  the  quotient  of  the  nume- 
rator divided  by  the  denominator.  (Art.  140.)  When  this 
quotient  can  not  be  expressed,  in  a  limited  number  of  terms, 
it  may  be  represented  by  an  infinite  series. 


296  INFINITE      SERIES. 


Ex.  1.  To  reduce  the  fraction to  an  infinite  series 

\—a 

divide  1  by  1— a,  according  to  the  rule  in  Art.  126. 

The  quotient  is  l+a+a2-fa3+a4+a5+fl6,  &c.  which 
shows  that  the  series,  after  the  first  term,  consists  of  the 
powers  of  a,  rising  regularly  one  above  another. 

That  the  series  may  converge,  that  is,  come  nearer  and 
nearer  to  the  exact  value  of  the  fraction,  it  is  necessary  that 
the  first  term  of  the  divisor  be  greater  than  the  second.  In 
the  example  just  given,  1  must  be  greater  than  a.  For  at 
each  step  of  the  division,  there  is  a  remainder;  and  the  quo- 
tient is  not  complete,  till  this  is  placed  over  the  divisor  and 
annexed.  Now  the  first  remainder  is  a,  the  second  «2,  the 
third  a3,  &c.  If  a  then  is  greater  than  1,  the  remainder  con- 
tinually increases;  which  shows,  that  the  farther  the  division 
is  carried,  the  greater  is  the  quantity,  either  positive  or  nega- 
tive, which  ought  to  be  added  to  the  quotient.  The  series  is, 
therefore,  diverging  instead  of  converging. 

But  if  a  be  less  than  1,  the  remainders,  a,  a2,  a3,  &c.  will 
continually  decrease.  For  powers  are  raised  by  multiplica- 
tion ;  and  if  the  multiplier  be  less  than  a  unit,  the  product 
will  be  less  than  the  multiplicand.  (Art.  85.)  If  a  be  taken 
equal  to  -J,  then  by  Art.  237, 

««w^  a3=l-,  a4  =  TY,  a5  =  ^,&,c. 
and  we  have 

T31  =  J=2=1+*+HHtV+3V+6V5&c. 

Here  the  two  first  terms    =  1+J,  which  is  less  than  2,  by  \\ 
the  three  first  =1+|,  less  than  2,  by  \; 

the  four  first  =  1  +&  less  than  2,  by  J ; 

So  that  the  farther  the  series  is  carried,  the  nearer  it  ap- 
proaches to  the  value  of  the  given  fraction,  which  is  equal  to  2. 

2.  If  — —  be  expanded,  the  series  will  be  the  same  as  that 

from >  except  that  the  terms  which  consist  of  the  odd 

powers  of  a  will  be  negative. 

So  that  — —  =1  —  fl+a2—  a3+aA~a5 +a°  &c. 
l+a 


INFINITE      SERIES.  297 


3.  Reduce  =■  to  an  infinite  series. 

a— b 

(h     bh      b2h    e 
a-b)h  (-  +  —  +  — >&c. 

'  \a     a2       a3 

7      bh 
a 


*     hh    * 
—  >  &c. 
a 

If  the  operation  be  continued  in  the  same  manner,  we 
shall  obtain  the  series, 

h     bh      b2h     b3h     b*h     0 
-  +  —  +  —  +  -7+  — r>  &c. 
a      a2       a3        a4        a5 

in  which  the  exponents  of  b  and  of  a  increase  regularly  by  1. 

4.  Reduce to  an  infinite  series. 

1  — a 

Ans.  l+2a+2a2+2a3+2a4,  &c. 

♦71*1.  Another  method  of  forming  an  infinite  series  is,  by 
extracting  the  root  of  a  compound  surd. 


Ex.  1.  Reduce  y/  a2  -\-b2  to  an  infinite  series,  by  extracting 
the  square  root  according  to  the  rule  in  Art.  508. 

,    /       b2        b*  b* 

a2+b2[a+- ■^-r  +  T7r-s>  &c. 

\       2a      8a3       16a5 


b2\  *      b* 

2a+2aJ 

°* 

b2J{ 

^4a2 


b2        b*  \  b* 

2a+ * ,  &c. 

a       8a3/  4a2 


8a3/  4a2 

b2        b*  b6 


2.   Va2—b2=a—- ,  &c. 

2a      8a3       16a  * 


3.    v/2=\/l  +  l  =  l+i~i+TV,  &c. 


x      x2      x3      5x* 


4.  vi+x=i+--T+--~,&c. 


298  INFINITE      SERIES. 

ol4.  A  binomial  which  has  a  negative  or  fractional  ex- 
ponent, may  be  expanded  into  an  infinite  series  by  the  bino- 
mial theorem.  See  Arts.  503,  505,  and  the  examples  at  the 
end  of  Sec.  xvi. 

Indeterminate  Co-efficients. 

o  1  •>.  A  fourth  method  of  expanding  an  algebraic  expres- 
sion, is  by  assuming  a  series,  with  indeterminate  co-efficients ; 
and  afterwards  finding  the  value  of  these  co-efficients. 

If  the  series,  to  which  any  algebraic  expression  is  assumed 
to  be  equal,  be 

A+Bx+Cx2+Dx3+Ex\  &c. 
let  the  equation  be  reduced  to  the  form  in  which  one  of  the 
members  is  0.  (Art.  183.)  Then  if  such  values  be  assigned 
to  A,  B,  C,  &c.  that  the  co-efficients  of  the  several  powers 
of  x,  as  well  as  the  aggregate  of  the  terms  into  which  x  does 
not  enter,  shall  be  each  equal  to  0 ;  it  is  evident  that  the 
whole  will  be  equal  to  0,  and  that,  upon  this  condition,  the 
equation  is  correctly  stated. 

The  values  of  A,  B,  C,  &c.  are  determined,  by  reducing 
the  equations  in  which  they  are  respectively  contained. 

a 

Ex.  1.   Expand  into  a  series        ,    » 

Assume  -^T-=A+Bx+Cx2+Dx:i+Ex*y  &c. 
c+ox 

Then  multiplying  by  the  denominator  c+bx,  and  trans- 
posing a,  we  have 

0=(Ac-a)  +  (Ab+Bc)x+(Bb+Cc)x*  +  (Cb+Dc)x*,  &c. 

Here  it  is  evident,  that  if  (Ac— a),  (Ab+Bc),  (Bb+Cc), 
&c.  be  made  each  equal  to  0,  the  several  parts  of  the  second 
member  of  the  equation  will  vanish,  (Art.  10G,)  and  the  whole 
will  be  equal  to  0,  as  it  ought  to  be,  according  to  the  assump- 
tion which  has  been  made. 

Reducing  the  following  equations, 

Ac— a=0,  we  have     A=  -> 

c 

Ab+Bc=0,  B=--A, 

c 

Bb+Cc=0,  C=--B, 


INFINITE      SERIES.  299 

Cb+De=0,  D=-h-Cy 

&c.  &c. 

That  is,  each  of  the  co-efficients,  C,  D,  and  E,  is  equal  to 

the  preceding  one  multiplied  into •     We  have  therefore, 

a  a      ab        ab2  ab3  ab* 

—x-\ -x2 -x3+—x\  &c. 


c-\-bx       c       c2  c3  c4  c 

a  +  frr 


2.  Expand  into  a  series 


d+hx  +  cx2 


a~\~bx 
Assume  ttt — ; o  =.A+ifo+C:c2-f-Ifo3,  &c. 

Then  multiplying  by  the  denominator  of  the  fraction,  and 
transposing  a-j-bx,  we  have  0=(Ad—  a)-\-(Bd+Ah  —  b)x 
+  (Cd+Bh+Ac)x2+(Dd+Ch+Bc)x3,  &c. 


erefore, 

A        a 
A=d' 

c—-dB-CdA' 

h  ,     6 

B=-dA+r 

B=-\C-\B. 

a+bx 

-a-    (h-A   l\,. 

_/*R4.-4U     k.o. 

d+hx-\-cx2     a     \a        a/        \a        a    / 

l+2;£ 

3.  Expand  into  a  series     _    __~j* 

Ans.  l+3^+4x2+7x3  +  llx4  +  18x5+29x«,  &c. 

In  which,  the  co-efficient  of  each  of  the  powers  of  x,  is  equal 
to  the  sum  of  the  co-efficients  of  the  two  preceding  terms. 

4.  Expand  into  a  series  ^ • 

r  b—ax 

61       ax      a2x2      a3x3      a*x*     ,     \ 

1 X 

5.  Expand  into  a  series   - — — -  • 

1        iCX       *jX 

Ans.   l+x+5^24-132:3+41x4+121x5+365a;6,  &c. 

6.  Expand  into  a  series - . 

r  1—  x— x2 +x3 

Ans.  l+x+2x2+2x3+3x*+3xs+4x0+4x\  &c. 


300  INFINITE      SERIES. 

l-X 


7. 


Expand  - — r--  8.  Expand  — 


-5x+6x2 


_  .       a+bx  1+x 

9.  Expand  — — ~Ty^'  10-  Jbxpand 


(1-dxy  xy"  WV"  (l-ar)»' 

In  the  preceding  examples,  the  series  assumed  contains  x 
and  its  powers,  with  positive  exponents,  rising  regularly  from 
the  second  term.  But  this  form  is  not  universally  applicable. 
There  are  some  algebraic  expressions  to  which  it  can  not  be 
applied,  without  leading  to  absurd  results ;  shewing  that  the 
method  is  not  fitted  to  expand  correctly  these  particular 
results. 

Summation  of  Series. 

510.  Though  an  infinite  series  consists  of  an  unlimited 
number  of  terms,  yet,  in  many  cases,  it  is  not  difficult  to  find 
what  is  called  the  sum  of  the  terms;  that  is,  a  quantity  which 
differs  less,  than  by  any  assignable  quantity,  from  the  value 
of  the  whole.  This  is  also  called  the  limit  of  the  series. 
Thus  the  decimal  0.33333,  &c.  may  come  infinitely  near  to 
the  vulgar  fraction  ^,  but  never  can  exceed  it,  nor,  indeed, 
exactly  equal  it.  See  Arts.  463,  4.  Therefore  \  is  the  limit 
of  0.33333,  &c.  that  is,  of  the  series 

_3_4-_3_4_ 3 4_ 3 I 3 £rC 

10     '     100     I     1000     '     I0OO0TlO0O00'    lx^" 

If  the  number  of  terms  be  supposed  infinitely  great,  the 
difference  between  their  sum  and  J,  will  be  infinitely  small. 

«> 17*  The  sum  of  an  infinite  series  whose  terms  decrease 

by  a  common  divisor,  may  be  found,  by  the  rule  for  the  sum 

of  a  series  in  geometrical  progression.     (Art.  452.)     Accord- 

rz  —  a 
ing  to  this,  $=  >  that  is,  the  sum  of  the  series  is  found 

by  multiplying  the  greatest  term  into  the  ratio,  subtracting 
the  least  term,  and  dividing  by  the  ratio  less  1.  But,  in  an 
infinite  series  decreasing,  the  least  term  is  infinitely  small. 
It  may  be  neglected  therefore  as  of  no  comparative  value. 
(Art.  466.)     The  formula  will  then  become, 

rz-0 

o= or  S=- 


r-1  r-1 

Ex.  1.  What  is  the  sum  of  the  infinite  series 

_3__l__3__l         _3 I 3 I     _         3  Arn 

10     »      100     '      1000     r10000T100«0fl)     VX,^. 


INFINITE      SERIES.  301 


Here  the  first  term  is  T\,  and  the  ratio  is  10. 

Then  S=— j  =   1Q_1  =f=l,  the  answer. 

2.  What  is  the  sum  of  the  infinite  series 

rz         2  V  1 

l  +  i+i+*  +  A  +  »V+»V.&c.?     Ans.  S= -  =  -±±=2. 

1 — 1       2—1 

3.  What  is  the  sum  of  the  infinite  series 

l  +  i+i+2T+eV  &c?  Ans.  f=l  +  f 

Recurring  Series. 

.  518.  When  a  series  is  so  constituted,  that  a  certain  num- 
ber of  contiguous  terms,  taken  in  any  part,  of  the  series,  have 
a  given  relation  to  the  term  immediately  succeeding,  it  is 
called  a  recurring  series;  as  any  one  of  the  following  terms 
may  be  found,  by  recurring  to  those  which  precede. 

Thus  in  the  series     l+3#+4x2+7:r3  +  ll:c4  +  18:c5,  &c., 
the  sum  of  the  co-efficients  of  any  two  contiguous  terms,  is 
equal  to  the  co-efficient  of  the  following  term. 

In  the  series         l+2x+3x2  +4x3  +5x*  +6x5 ,  &c, 
each  term,  after  the  second,  is  equal  to  2x  multiplied  by  the 
term  immediately  preceding,  —  x2  multiplied  by  the  term  next 
preceding.     The  co-efficients  of  x  and  x2,  that  is,  +2—1, 
constitute  what  is  called  the  scale  of  relation. 

In  the  series  l+4x+6^2  +  ll^3+28a:4+63^5,  &c, 
any  three  contiguous  terms  have  a  constant  relation  to  the 
succeeding  term.  The  scale  of  relation  is  2—1+3;  so  that 
each  term,  after  the  third,  is  equal  to  2x  into  the  term  imme- 
diately preceding,  —  x2  into  the  term  next  preceding,  +3.r3 
into  the  third  preceding  term. 

Let  any  recurring  series  be  expressed  by 
A+B+C+D+E+F,  &c. 

If  the  law  of  progression  depends  upon  two  contiguous  terms 
and  the  scale  of  relation  consists  of  two  parts,  m  and  n, 

Then     C—Bmx-\-Anx2,  the  third  term, 
D=Cmx+Bnx2,  the  fourth. 
E=Dmx+Cnx2,  the  fifth, 
&c.  &c. 

26 


302  INFINITE      SERIES. 

If  the  law  of  progression  depends  on  three  contiguous 
terms,  and  the  scale  of  relation  is  m-\-n+r, 

Then     D—Cmx  +  Bnx2-\-Arx3,  the  fourth  term, 
E=Dmx+Cnx2+Brx3,   the  fifth, 
F=Emx+D?ix2  +  Crx3,   the  sixth, 
&c.  &c. 

If  the  law  of  progression  depends  on  more  than  three  terms, 
the  succeeding  terms  are  derived  from  them  in  a  similar 
manner. 

•519.  In  any  recurring  series,  the  scale  of  relation,  if  it 
consists  of  two  parts,  may  be  found,  by  reducing  the  equa- 
tions expressing  the  values  of  two  of  the  terms ;  if  it  con- 
sists of  three  parts,  it  may  be  found  by  reducing  the  equations 
expressing  the  values  of  three  terms,  &c.  As  the  scale  of 
relation  is  the  same,  whatever  be  the  value  of  x  in  the  series, 
the  reduction  may  be  rendered  more  simple,  by  making  x=l. 

Taking  then  the  fourth  and  fifth  terms,  in  the  first  example 
above,  and  making  x—l,  we  have 

t?__t)     ,  ^    C    to  find  the  values  of  m  and  n. 

These  reduced,  give 

DC-BE  CE-DD 

CC-BD  n~CC-BD* 

A    B      C  D       E        F 

1  +3x+5x2  +7x*  +9x*  +Ux5 ,  &c. 
Making  x=l,  we  have 

7X5-3X9  5X9-7* 

Therefore,  the  scale  of  relation  is  2—1. 

To  know  whether  the  law  of  progression  depends  on  two, 
*thme,  or  more  terms  ;  we  may  first  make  a  trial  of  two  terms ; 
and  if  the  scale  of  relation  thus  found,  does  not  correspond 
with  the  given  series,  we  may  try  three  or  more  terms.  Or 
if  we  begin  with  a  number  of  terms  greater  than  is  neces- 
sary, one  or  more  of  the  values  found  will  be  0,  and  the 
others  will  constitute  the  true  scale  of  relation. 

o20.  When  the  scale  of  relation  of  a  decreasing  recur- 
ring series  is  known,  the  sum  of  the  terms  may  be  found. 


m=  j^ 

In  the  series 


INFINITE      SERIES.  303 


Let      \A 

I  a- 


be  a 


+bx+cx2  +dx3  +ex*  +fx5 ,  &c. 
be  a  recurring  series,  of  which  the  scale  of  relation  is  m+n: 

Then     A=  the  first  term,         J5=  the  second, 
C=Bxmx+Axnx2,  the  third, 
D=Cxmx+Bxnx2,  the  fourth, 
E=Dxmx+Cxnx2,  the  fifth. 
&c.  &c. 

Here  mx  is  multiplied  into  every  term,  except  the  first  and 
the  last;  and  nx2  into  every  term  except  the  two  last.  If 
the  series  be  infinitely  extended,  the  last  terms  may  be  neg- 
lected, as  of  no  comparative  value,  (Art.  466,)  and  if  S=  the 
sum  of  the  terms,  we  have 

S=A+B+mzX(B+C+D,  &c.)+nx2  X(A+B+C,  &c.) 
But  S-A=B+C+D,  &c.      And  S=A+B+C,  &c. 

Therefore  S=A+B+mxX(S-A)+nx2  xS. 

Reducing  this  equation,  we  have 

A+B—Amx 

&=  -. ^  • 

l  —  mx—nx2 

Ex.  1.  What  is  the  sum  of  the  infinite  series 

l+6x  +  12x2+48x3  +  120^4,  &c? 

The  scale  of  relation  will  be  found  to  be  1+6. 

Then     A=l,         B=6x,         m=l,         n=6. 

I  _|_  5  x 

The  series  therefore  =  : ~z* 

l—x  —  6x2 

2.  What  is  the  sum  of  the  infinite  series 
l+3a:+4^2+7x3  +  lla:4  +  18a:5+29a?6,&c.?     Ans. 


1-x-x2 

3.  What  is  the  sum  of  the  infinite  series 

l+x+5x2  +  13x3  +4lx*  +I2lx5  +365x*,  &c.  ? 

4.  What  is  the  sum  of  the  infinite  series 

l+2x— 2x  1 


l+2x+3o;3-h4.r3-h5^*,&c.?        Ans. 


l-2x+#2       (1—  xY 


304  INFINITE      SERIES. 


Method  of  Differences. 

•721.  In  the  Summation  of  Series,  the  object  of  inquiry 
is  not,  always,  to  determine  the  value  of  the  whole  when  in- 
finitely extended  ;  but  frequently,  to  find  the  sum  of  a  certain 
number  of  terms.  If  the  series  is  an  increasing  one,  the  sum 
of  all  the  terms  is  infinite.  But  the  value  of  a  limited  num- 
ber of  terms  may  be  accurately  determined.  And  it  is  fre- 
quently the  case,  that  a  part  of  a  decreasing  series,  may  be 
more  easily  summed  than  the  whole.  A  moderate  number 
of  terms  at  the  commencement  of  the  series,  if  it  converges 
rapidly,  may  be  a  near  approximation  to  the  amount  of  the 
whole,  when  indefinitely  extended. 

One  of  the  methods  of  determining  the  value  of  a  limited 
number  of  terms,  depends  on  finding  the  several  orders  of 
differences  belonging  to  the  series.  The  differences  between 
the  terms  themselves,  are  called  the  first  order  of  differences  ; 
the  differences  of  these  differences,  the  second  order,  &c.  In 
the  series, 

1,  8,  27,  64,  125,  &c. 
by  subtracting  each  term  from  the  next,  we  obtain  the  first 
order  of  differences, 

7,  19,  37,  61,  &c. 
and  taking  each  of  these  from  the  next,  we  have  the  second 
order, 

12,  18,  24,  &c. 

Proceeding  in  this  manner  with  the  series 
a,  b,  c,  d,  e,  f3  &c. 
we  obtain  the  following  ranks  of  differences, 

1st.  Diff.    b— a,  c—b,  d—c,  e—d,  f—e,  &c. 

2d.  Diff.    c-2b+ a,  d-2c+b,  e-2d+c,  f-2e+d,  &c. 

3d.  Diff.    d-3c+3b-a,  e-3d+3c-b,  f-3e+3d-c,  &c. 

4th.  Diff:    e-4d+C)C-4b+a,  f-4e+6d-4c+b,  &c. 

5th.  Diff.    f-5e+l0d-10c+5b-a,  &c. 

&c  &c. 

.  In  these  expressions,  each  difference,  here  pointed  off  by 
commas,  though  a  compound  quantity,  is  called  a  term.    Thus 
the  first  term  in  the  first  rank  is  b—a  ;  in  the  second,  c-2b-{-a; 
in  the  third,  d—  3c-\  3b  —  a,  &c.      The  first  terms,  in  the  3cv 
era]   orders,   are  those  which  are  principally  employed   in 


INFINITE      SERIES.  305 

investigating  and  applying  the  method  of  differences.  It 
will  be  seen,  that  in  the  preceding  scheme  of  the  successive 
differences,  the  co-efficients  of  the  first  term, 

In  the  second  rank,  are  1,      2,      1 ; 

In  the  third,  1,      3,      3,      1 ; 

In'the  fourth,  1,      4,      6,      4,      1 ; 

In  the  fifth,  1,  -  5,     10,    10,     5,      1 ; 

Which  are  the  same,  as  the  co-efficients  in  the  powers  of 
binomials.  (Art.  488.)  Therefore,  the  co-efficients  of  the  first 
term  in  the  nth  order  of  differences,  (Art.  493,)  are 

71—1  n—1      71  —  2      0 

1,  n>  7^x"~^~,  nX~2~ X~3~~ '     c# 

522.  For  the  purpose  of  obtaining  a  general  expression 
for  any  term  of  the  series  a,  b,  c,  d,  &c.  let  D',  D",  Z>",  D"", 
&c.  represent  the  first  terms,  in  the  first,  second,  third,  fourth, 
&c.  orders  of  differences. 

Then     D'    =b-a, 

D"  =c-2b+a, 
D"  =d-3c+3b-a, 
D'o^e-ld+dc-ib-ha, 
&c.  &c. 

Transposing  and  reducing  these,  we  obtain  the  following 
expressions  for  the  terms  of  the  original  series,  a,  b}  c,  d,  &c. 

The  second  term  b=a-{-D', 

The  third,  c=a  +2D+D", 

The  fourth,  d=a+3D,+3D"+D'", 

The  fifth,  e=a+4Df+6-D"+4D'"+D"". 

Here  the  co-efficients  observe  the  same  law,  as  in  the 
powers  of  a  binomial;  with  this  difference,  that  the  co- 
efficients of  the  Tith  term  of  the  series,  are  the  co-efficients 
of  the  (n  —  l)th  power  of  a  binomial. 

Thus  the  co-efficients  of  the  fifth  term  are  1,  4,  6,  4,  1 ; 
which  are  the  same  as  the  co-efficients  of  the  fourth  power 
of  a  binomial.  Substituting,  then,  n-l  for  n,  in  the  formula 
for  the  co-efficients  of  an  involved  binomial,  (Art.  493,)  and 
applying  the  co-efficients  thus  obtained  to  B\  D",  D",  D"", 
&c.  as  in  the  preceding  equations,  we  have  the  following  gen- 
eral expression,  for  the  nth  term  of  the  series,  a,  b,  c,  d,  &c. 

26* 


306  INFINITE      SERIES. 

The  nth  term 

When  the  differences,  after  a  few  of  the  first  orders,  become 
0,  any  term  of  the  series  is  easily  found. 

Ex.  1.  What  is  the  nth  term  of  the  series  1,  3,  6, 10, 15,  21  ? 
Proposed  series,  1,  3,  6,  10,  15,  21,  &c. 

First  order  of  differences,      2,  3,  4,  5,  6,   &c. 
Second     do.  1,  1,  1,  1,   &c. 

Third     do.  0,  0,  0. 

Here  «=1,     Z>'=2,     D"=l,     Z)'"=0. 

•  n 2 

Therefore  the  nth  term  =  l  +  (rc-l)2+(tt-l)— — 

m 

The  20th  term  =1+38+171=210.     The  50th  =1275. 

2.  What  is  the  20th  term  of  the  series  1 3,  23,  33,  43,  53,  &c? 
Proposed  series,  1,     8,     27,     64,     125,  &c. 

First  order  of  differences,    7,     19,     37,     61,  &c. 
Second     do.  12,     18,     24,  &c. 

Third     do.  6,       6,   &c. 

Here   D'=7,         Z>"=12,         D'"=6. 
Therefore  the  20th  term  =8000. 

3.  What  is  the  12th  term  of  the  series  2,  6,  12,  20,  30, 
&c?  Ans.  156. 

4.  What  is  the  15th  term  of  the  series  l2,  22,  3a,  4a,  52,  62, 
&c?  Ans.  225. 

*>23.  To  obtain  an  expression  for  the  sum  of  any  number 
of  terms  of  a  series  a,  b,  c,  d}  &c.  let  one,  two,  three,  &c.  terms 
be  successively  added  together,  so  as  to  form  a  new  series, 
0,     a,     a+b,     a+b+c,     «+^+c+rf,  &c. 

Taking  the  differences  in  this,  we  have 
1st  Diff.         «,         b,         c,         d,        e,        f,  &c. 
2d    Diff.    b—a,  c—by  d—c,  e—d,  f—e,  &c. 
3d    Diff.    c-2b+a,  d-2c+b,  e-2d+c,  f-2e+d,  &c. 
4th  Diff.    d-3c+3b— a,  e—3d+3c—b,  f—3e+3d-c,  &c. 
&c.  &c. 


INFINITE      SERIES.  307 

Here  it  will  be  observed  that  the  second  rank  of  differences 
in  the  new  series,  is  the  same  as  the  first  rank  in  the  original 
series  a,  b,  c>  d,  e}  &c.  and  generally,  that  the  (n  +  l)th  rank 
in  the  new  series  is  the  same  as  the  7ith  rank  in  the  original 
series.  If,  as  before,  D'=  the  first  term  of  the  first  differ- 
ences in  the  original  series,  and  d'=  the  first  term  of  the  first 
differences  in  the  new  series ; 

Then        d'=a,      *'=*&)      d'"=D",      d""=D'"f  &c 

Taking  now  the  formula  (Art.  522,) 

a+(n^l)D'  +  (n-l)^D'f  +  (n-l)^X7^D/ff+  &c. 

which  is  a  general  expression  for  the  nth  term  of  a  series  in 
which  the  first  term  is  a ;  applying  it  to  the  new  series,  in 
which  the  first  term  is  0,  and  substituting  n-f-1  for  n,  we  have 

71-1   :  71-1       71-2  ,  71-1       71-2      71-3  , 

0+nd'+n—-  d"+n—X  —  d"'+n—-X  —  X—rd'"'+&c. 

Z  Z  o  Z  o  4 

^  71-1^  71-1       7Z-2^  71-1       71-2      n-S  _ 

Or  na+n-—D'  +  n—-X -^-D,f+n-—  X~^~X  -—D'"+&c. 

Z  Z  o  Z  o  4 

Which  is  a  general  expression  for  the  (n-fl)th  term  of  the  series 

0,     a,     a-\-b,     a-{-b+c,    .a  +  b+c+d,  &c. 
or  the  Tzth  term  of  the  series 

a,     a  +  b,     a-\-b-\-c,     a+b+c+d,  &c. 

But  the  Tith  term  of  the  latter  series,  is  evidently  the  sum 
of  n  terms  of  the  series,  a,  b,  c,  d,  &c.  Therefore  the 
general  expression  for  the  sum  of  n  terms  of  a  series  of  which 
a  is  the  first  term,  is 

71-1  72-1       7^-2^  71-1       7Z-2       71-3^ 

na+n-—  D'+n—  X-^D,'-\-n—-X-7rX—r-D'"Jr  &c. 

Z  Z  o  Z  o  4 

Ex.  1.  What  is  the  sum  of  n  terms  of  the  series  of  odd 
numbers,  1,  3,  5,  7,  9,  &c.  ? 

Series  proposed,  1,  3,  5,  7,  9,  &c. 

First  order  of  difference,         2,  2,  2,  2,    &c. 
Second  order  of  difference,       0,  0,  0. 

Here    a=\3  D=2,  D"=0. 

n— -1 

Therefore  the  sum  of  n  terms  =71+71  — —  X2=n2. 

z 

That  is,  the  sum  of  the  terms  is  equal  to  the  square  of  the 
number  of  terms.     See  Art.  441. 


308  INFINITE      SERIES. 

2.  What  is  the  sum  of  n  terms  of  the  series 

l2,  22,  32,  42,  52,  &c? 
Here  a=l,         D'=3,         Z>'=2,         D'"=0. 
Therefore  n  terms  =  }(2n3+3n2 +n)  =  }n(n  +  l)X(2n  +  l). 
Thus  the  sum  of  20  terms  =2870. 

3.  What  is  the  sum  of  n  terms  of  the  series 

l3,  23,  33,  43,  &c? 

Here  a=l,     D'=7,     D"=12,     D"=6,     D'"=0. 

Therefore  n  terms  =\(n*  +2n3+n2)  =  (^nXn+l)2 . 
Thus  the  sum  of  50  terms  =1625625. 

4.  What  is  the  sum  of  n  terms  of  the  series 

2,  6,  12,  20,  30,  &c?         Ans.  %n(n  +  l)x(n+2). 

5.  What  is  the  sum  of  20  terms  of  the  series 

1,  3,  6,  10,  15,  &c? 

6.  What  is  the  sum  of  12  terms  of  the  series 

l4,  24,  34,  44,  54,  &c.? 

Continued   Fractions. 

524.  One  form  of  series  is  given  in  the  algebraic  expres- 
sions which  are  denominated  continued  fractions.  Among 
the  uses  to  which  these  are  applied,  one  of  the  principal  is  in 
approximating  to  the  values  of  fractions  which  are  expressed 
in  large  numbers.     If  for  the  purpose  of  reducing  the  frac- 

.       1393        .  „  ..,.    .    _  .    .,     - 

tion  -— — -  to  lower  terms,  we  divide  it  by  the  denominator, 
972  J 

we  obtain  the  quotient  1,  and  a  remainder,  which  placed 

over  the  divisor,  gives  the  traction  — — - 

°  0  972 

130 

Dividing  by  the  numerator,  we  have  -J-+  the  fraction  — - 

31 

Dividing  again,  we  have  ^+  the  fraction  -  — - 

&c.  &c. 

Here,  each  of  the  fractions,  after  the  first,  is  divided  by  its 
numerator,  which  is  the  remainder  in  the  preceding  divis- 
ion ;  the  preceding  divisor  now  becoming  a  dividend,  as  in 
the  process  for  finding  the  greatest  common  measure  of  two 
quantities,  (Art.  476.)     Each  of  the  denominators,  after  the 


INFINITE      SERIES.  300 

division,  consists  of  two  parts,  an  integer  and  a  fraction  less 
than  a  unit.     The  results  may  be  arranged  as  follows. 

1393-7  ,  l     7      Generally,  «        , I     ,     Or  ?__i     , 

6  &C-  &c. 

If  the  numerator  of  the  original  fraction  is  greater  than 
the  denominator,  the  series  will  begin  with  an  integer,  as  in 
the  first  two  forms  above.  But  if  the  numerator  is  less  than 
the  denominator,  the  series  will  begin  with  a  fraction,  as  in 
the  third  form. 

«?3t>.  A  continued  fraction,  then,  is  one  whose  denomina- 
tor is  a  whole  number  and  a  fraction ;  the  denominator  of 
the  latter  being  also  a  whole  number  and  a  fraction ;  the 
formula  being  continued,  in  a  series  of  similar  fractions. 

To  throw  a  common  fraction  into  the  form  of  a  continued 
fraction,  we  have,  then,  the  following  rule ;  Divide  the  greater 
term  of  the  given  fraction  by  the  less,  and  this  divisor  and 
the  following  ones,  by  the  several  remainders  successively ; 
as  in  finding  the  greatest  common  measure  of  two  numbers. 

The  fractions  ->  -»  -»  &c.  taken  by  themselves  are  called 
p     q     r  * 

integral  fractions,  as  their  denominators  are  whole  numbers, 

(Art.  524.)     Each  integral  fraction  is  added,  not  to  the  whole 

preceding  one,  but  to  its  denominator  only. 

536.  The  separate  expressions  -#  -  .  1    —  .  1     , 
r  p    p+~,  p.] —     i 

r    r     q    r     q+~,  &c. 
r 

are  called  approximating  or  converging  fractions,  as  each,  in 

succession,  gives  a  nearer  approximation  than  the  preceding 

one,  to  the  exact  value  of  the  original  fraction  r  •    They  are 

not  independent  expressions,  as  each  following  one  is  only 
an  extension  of  the  preceding  one.     In  the  formula  already 

1  a  I 

given,  -     1  is  nearer  the  value  of  t  than  - .     For  an  addi- 

jp+-  b  p 

Hon  must  be  made  to  the  denominator  of  ->  to  render  it 


310  INFINITE      SERIES. 

a 
equivalent  to  r-    (Art.  524.)    And  a  'part  of  this  addition  is 

made  in  -     1     Again,  the  third  converging  fraction  -     1 

9  P    9+' 

r 

a  1 

is  nearer  in  value  to  r  >  than  the  second  -  ,  1      For  an  addi- 
b  p+-' 

9 

tion  must  be  made  to  the  denominator  of  this,  to  render  it 

a 
equivalent  to  r  J    and  a  part  of  this  addition  is  made  in 

-     1     -     The  same  proof  may  be  extended  to  each  succes- 

q+— 

J     r 

sive  converging  fraction.  That  portion  of  the  original  quan- 
tity which  remains  unexhausted  by  the  continued  fraction  is 
diminished,  by  each  successive  addition  of  an  integral  fraction. 

527.  The  first  converging  fraction  -   exceeds  the  orig- 

a 
inal  t*     For  the  denominator  p  is  too  small,  and  this  ren- 
ders the  value  of  the  fraction  too  great.  (Art.  143,  cor.)   But 
the  second  converging  fraction  -     1  is  less  than  r>   for  the 

P    9 

denominator  q  is  too  small,  and  this  renders  ->  and  of  course 

9 

p-\ —  too  great;   and  therefore  the  value  of  -  ,1    too  small. 

rq       &  p+- 

q 

In  the  same  manner,  it  may  be  shown,  that  the  third  con- 
verging fraction  is  too  great,  the  fourth  too  small,  &c.  that 
is,  Of  several  successive  converging  fractions,  those  which 
are  counted  by  odd  numbers  are  too  large,  while  those  which 
are  counted  by  even  numbers  are  too  small.  Each  succeed- 
ing one,  however,  comes  nearer  than  the  preceding  one,  to 
the  value  of  the  original  fraction.  It  follows,  that  the  exact 
value  of  this  is  between  two  consecutive  fractions,  one  of 
which  is  too  great,  and  the  other  too  small. 

«>28.   When  the  numerator  and  denominator  of  the  given 
fraction  are  commensurable,  the  series  of  continued  fractions 


INFINITE      SERIES.  311 

derived  from  it  will  terminate;  as  the  process  of  forming  it, 
after  several  divisions,  will  give  a  quotient,  without  a  remain- 
der. But  if  the  terms  of  the  original  fraction  are  incommen- 
surable, the  divisions  may  be  continued  without  limit,  forming 
an  infinite  series. 

Ex.  1.  What  is  the  continued  fraction  corresponding  to 
235  .      .  .  11111 

Ans.  1  he  successive  integral  fractions  are 


1683  &  7  6  5  2  3 

The  last  of  these  leaves  no  remainder.  The  series,  therefore, 
terminates  here ;  and  the  last  continued  fraction  is  exactly 
equal  to  the  vulgar  fraction  from  which  it  is  derived. 

2.  What  is  the  continued  fraction  formed  from  ? 

887 

Ans.  The  integral  fractions  are  l+-r>  ->  ->  ->  t<  -• 

4    9    2    114 

3.  What  are  the  integral  fractions  derived  from   — —  ? 

11111 

Ans.  24  +  r  -,  j.  -,  -• 

•529.  The  values  of  other  quantities  besides  fractions, 
particularly  of  surd  quantities,  may  be  expressed  in  the  form 
of  continued  .fractions.  If  x  be  a  quantity  which  can  not  be 
exactly  stated  in  whole  numbers,  let  the  greatest  integer  con- 
tained in  it  be  a.     Then  x~a-\-  a  fraction  less  than  a  unit. 

Let  the  latter  be  -•     Its  denominator  y  must  be  greater 

than  1.    (Art.  141.)     Let  the  greatest  integer  contained  in  it 

be  b,  and  the  remaining  fraction  be   -r-     Then  y=b+  —  • 

°  y  y 

In  like  manner,  let  the  greatest  integer  contained  in  y'  be 
b',  and  the  remaining;  fraction  be  —  >  &c.     We  have  then 

y" 

x=a+-,  y=b+-r  y'=b'+\,  ^'=6"+t£7>  &c.       Then 

y  y  y  y 

substituting  for  y,  yf,  y",  &c.  their  several  values,  we  have 
I  1     ,  1     ,  1     , 

y  y"  6//+^/-*c 


312  INFINITE      SERIES. 

Ex.  The  square  root  of  2  is  1.414213+  in  decimals,  or 
14142134-        Tr    .  .    .      .. 

I  oooooo —  divided,  according  to  the  rule  in  Art. 

525,  the  first  six  integral  fractions  will  be  each  -V  forming 
the  continued  fraction   ^2=1+-     1     , 

•TJIO.  A  continued  fraction  may  be  reduced  to  a  vulgar 
fraction,  and  its  value  found,  by  the  common  rules  for  the 
reduction  of  fractions  ;  and  by  comparing  several  converging 
fractions,  a  rule  is  obtained,  by  which,  if  any  contiguous  ones 
in  the  series  are  known,  the  value  of  the  next  may  he  found. 
Of  the  three  following  converging  fractions, 

The  1st, 

II  1             ap  +  l 
=-                      Or    a-\ —        = 

P  P  P  P 

The  2d, 

1      ,  <7  v  1      ,       apq+a+q 

-     1       =-*—  (Art.  171.)  a+-,l  =-^-—--±- 

The  3d, 
I     1  qr+1  ill         _  (apq+a+q)r+ap  +  l 

*>+-+l~~(pq  +  l)r+p      a    P+~+l~~  (pq+l)r+p 

r  *     r 

Here  it  will  be  seen,  that  the  numerator  of  the  third  con- 
verging fraction  is  equal  to  the  product  of  the  numerator  of 
the  preceding  fraction,  into  the  denominator  of  the  third  inte- 
gral fraction,  added  to  the  numerator  of  the  first  converging 
fraction ;  and 

The  denominator  is  equal  to  the  product  of  the  denomina- 
tor of  the  preceding  fraction  into  the  denominator  of  the 
third  integral  fraction,  added  to  the  denominator  of  the  first 
converging  fraction. 

To  show  that  this  is  a  general  law,  let  the  three  fractions 
above  be  represented  by 

N    N'  N" 

V  &'  and  wm 

N"  =  iW-fiV 
Then,  as  has  just  been  shown,  «=        rkl    ,  ^. 

U    =  JJ  r-\-JJ 


INFINITE      SERIES  313 

N'" 
Let  the  fourth  converging  fraction  jj-  be  now  added,  its 

last  integral  fraction  being  -•     The  only  change  here  made, 

is  in  adding  -  to  r.     To  obtain  the  value,  then,  we  have 

only  to  substitute  r-\ —  for  r,  making 

jynJV^+^+N ^  (N'r+N)s+N> 

s 
N'r+N=N"      _,       ,       N'"  =  N"s+N' 
But  W7+B=W    Therefore  w  =W7+&' 

And  as  one  converging  fraction  after  another  is  formed,  by 
merely  adding  a  new  integral  fraction,  we  have  this  general 
law; 

Of  any  three  consecutive  converging  fractions,  the  nume- 
rator of  the  last  is  equal  to  the  product  of  the  numerator  of 
the  preceding  one,  into  the  denominator  of  the  last  integral 
fraction,  added  to  the  numerator  of  the  first  of  the  three;  and 

The  denominator  of  the  last  is  equal  to  the  product  of  the 
denominator  of  the  preceding  one,  into  the  denominator  of 
the  last  integral  fraction,  added  to  the  denominator  of  the 
first  of  the  three. 

531.  The  difference  of  the  numerators  of  two  contiguous 
converging  fractions  is  either  +1  or  —  1 ;  and  the  denomina- 
tor of  this  difference  is  the  product  of  the  denominators  of 
the  two  fractions. 

Thus    i-     q     -W+l-W L, 


And 


p    P9+1     p(pq+l)    p(pq+l) 

q  qr+l        _  —  1 


P9  +  l      (Pq  +  l)r+p      (pq+l)X(pq  +  l)r+p 

And  if  any  three  consecutive  converging  fractions  be 
taken,  the  difference  of  the  numerators  of  the  first  two,  when 
reduced  to  a  common  denominator,  is  the  same,  with  a  coiw 
trary  sign,  as  the  difference  of  the  numerators  of  the  other 
two. 

27    • 


814  INFINITE      SERIES. 

N  N'   N"    ' 
Let  -yy  -jy>  jy;>  be  three  consecutive  converging  fractions. 

N     N'      ND-N'D 
Then    ^  —  tt:  = 


And 


D      I)'  DD' 

N'      N"      N'D"-N"D' 


D'      D"  ~~        D'D" 

But  N"=N'r+N.     (Art.  530.)     And   D"=D'r+D. 

■     .      .         .         N'     N"     N'(D'r+D)-D'(N'r+lV) 
By  substitution,  then,   jy  —  jyr,  = JFD" 

mu.       u                              N'     N"      -ND'+N'D 
This,  when  reduced,  gives  jy~  jy,= Tyjy' 

Shewing  the  difference  between  the  last  two  numerators 
to  be  the  same,  with  the  contrary  sign,  as  the  difference  be- 
tween the  first  two. 

As  in  the  example  at  the  beginning  of  this  article,  the  dif- 
ference of  the  first  two  enumerators  is  +1,  and  the  difference 
of  the  last  two  is  —  1  ;  it  follows  from  this,  and  the  general 
law  now  proved,  that  the  difference  must  be  alternately  +1 
and  —I,  throughout  the  series. 

532.  The  excess  or  deficiency  of  any  converging  fraction 
is  less  than  a  unit  divided  by  the  square  of  its  denominator, 
when  reduced  to  a  vulgar  fraction. 

N    N'    N" 
Let  -j^>   -jj,>  -j^j>  be  three  contiguous  fractions,  and  the 

last  integral  fraction  be  ->  r  being  a  whole  number.  (Art.  525.) 

JV  N"  dzl 

The  difference  between  the  values  of  -^  and  -jt--  is 


D'  D"       D'D11' 

(Art.  531.)    As  the  value  of  the  original  quantity  from  which 
these  are  derived  is  between  the  two,  (Art.  527,)  it  follows. 

that  the  excess  or  deficiency  of  either  is  less  than  77,7^  • 

But  as  D"=D'r+D,  (Art.  530,)  D"  is  greater  than  D' ,  and 

of  course,  D'D"  is  greater  than  D'D'.     Therefore    ])t  J)f/  is 

less  than  g-j«    (Art.  143,  cor.)     And  as  the  excess  or  defi- 


INFINITE      SERIES.  315 

ciency  of  -yy  is  less  than  jyfj^>  stlu  more  is  it  less  than  yr^- 

As  the  denominators  of  the  continued  fractions  go  on  in- 
creasing, while  the  series  is  extended,  we  have  proof  here,  as 
well  as  in  Art.  526,  that  the  small  error  which  each  contains 
is  continually  diminishing ;  and  that  the  formulas  are  ap- 
proaching nearer  and  nearer  to  the  exact  value  of  the  quan- 
tity from  which  they  are  derived. 

533.  One  of  the  applications  of  the  principles  of  con- 
tinued fractions  is  in  finding  eligible  numbers  to  express 
nearly  the  ratio  of  the  diameter  of  a  circle  to  its  circumfer- 

314159 

ence.     This,  in  decimals,  is  3.14159+,  or  +.     If  we 

divide  according  to  the  rule  in  Art.  525,  we  obtain  the  follow- 

3     111 

ing  integral  fractions,  ->  ->   — »   -»  &c.    Reducing  the  con- 

1       7       io      1 

verging  fractions  formed  with  these,  (Art.  530,)  we  have  the 

<-n       :  i        «      ♦■  3     22     333     355    a         rr,, 

following  vulgar  tractions,  y*  —*  — —  >  iT^'  *  1  he  sec- 
ond of  these  gives  the  proximate  ratio  of  the  circumference 
to  the  diameter  22  to  7,  as  demonstrated  by  Archimedes, 
The  fourth  gives  it  much  more  accurately,  355  to  113,  the 
ratio  assigned  by  Metius. 

Interpolation. 

*>34.  Logarithms,  trigonometrical  sines,  tangents,  &c. 
are  given  in  the  tables,  for  the  natural  series  of  numbers,  or 
for  degrees,  minutes,  &c.  We  sometimes  have  occasion  to 
find  the  intermediate  logarithms,  sines,  &c.  for  fractional  parts 
of  a  degree,  unit,  &c.  This  is  called  interpolation.  The 
formula 

a  +  (n-l)D'  +  (n-l)X^Dl'  +  (n-l)x~xn^D'"+kc. 
in  Art.  522,  answers  this  purpose. 

When  the  terms  of  a  series  abed,  &c.  are  numbered 
12  3,  &c.  the  interval  between  each  of  the  numbers  and  the 
next  being  considered  a  unit;  it  is  evident  that  ?i —  1  ex- 
presses in  portions  of  this  interval,  the  distance  of  the  nth 


JD'=3091 

D"=-22 

3069 

3048 

-21 

316  INFINITE      SERIES. 

number  from  the  first.     If  then  m  be  substituted  for  n— I, 
the  formula  for  the  ?ith  term  becomes 

a+mD'+m.T^D"+m.^p  ~D"'+  &c. 

I£z.  1.  Given  the  logarithms  of  140,  141,  142,  143,  to  find 

the  logarithm  of  141  -• 

The  interval  here  between  the  numbers  is  1.  The  distance 
of  the  number  whose  logarithm  is  required  from  the  first  is 

1-=-.     The  logarithms  given  in  the  tables  are 

For  140,   146128 

141,  149219   ~   3-*  Dm=l 

142,  152288 

143,  155336 

As  the  differences  of  the  first  order  decrease,  while  the 
logarithms  from  which  they  are  derived  increase,  the  differ- 
ences of  the  second  order  are  negative.    (Art.  52.) 

The  first  term  a  =146128 

3 
The  second,  mD'=-X 3091  =     4636 

m-\  3     1  150764 

The  third,  m — — D'=-X-X-22  =       -8 

The  logarithm  required  is  150756 

In  this  example,  the  first  three  terms  of*  the  formula  give 
the  result  required,  with  a  sufficient  degree  of  exactness. 
And  generally,  a  few  terms  will  answer  the  purpose.  Almost 
all  the  logarithms  in  the  tables  are  approximations  only ;  being 
obtained  by  series  which  converge,  but  do  not  terminate. 

2.  Given  the  logarithmic  sines  of  20°,  '22*,  24°,  26°;  viz. 
534052,  573575,  609313,  641842,  to  find  the  sine  of  23£ 
degrees. 

Here  the  interval  between  the  numbers  is  2.  The  dis- 
tance m  of  the  required  sine  from  the  first  number  is  If  of 

•y  jyi \  7      3 

this  interval.     Therefore    mD'=-D',  m — -—D"=-X-D\ 


4     8 


m—1   m  —  2^       7     3  1 

-—•'■-  Y-D'"=lX8x-¥2D" 

The  sine  required  is  600700. 


GENERAL     PROPERTIES      OF     EQUITATIONS.       317 

3.  Given  the  logarithmic  tangents  of  17°,  18°,  19°,  20°; 
viz.  485339,  511776,  536972,  561066,  to  find  the  tangent  of 
18°£.  Ans.  524520. 

4.  Given  the  natural  sines  of  24°,  25°,  26°,  27° ;   vizr- 
40674,  42262,  43837,  45399,  to  find  the  natural  sine  of  25°| . 

Ans.  43313. 


SECTION    XIX. 

GENERAL    PROPERTIES    OF    EQUATIONS. 

Art.  t>3t>«  Equations  of  any  degree  may  be  produced 
from  simple  equations,  by  multiplication.  The  manner  in 
which  they  are  compounded  will  be  best  understood,  by  taking 
them  in  that  state  in  which  they  are  all  brought  on  one  side 
by  transposition.  (Art.  183.)  It  will  also  be  necessary  to 
assign,  to  the  same  letter,  different  values,  in  the  different 
simple  equations. 

Suppose,  that  in  one  equation,    x=2  ) 


And,  that  in  another, 

x=S  ) 

By  transposition, 

#-2=0 

And 

#-3  =  0 

Multiplying  them  together,  x2  —  5#+6=0 

Next,  suppose,  #—4=0 

And  multiplying,  x3  —  9#2  +26#  —  24- 

Again  suppose,  x  —  5=0 


And  mult,  as  before,  #4  — 14#3+71#2  — 154#+120=0,&c. 

Collecting  together  the  products,  we  have 
(#-2)(#-3)  =#2-5#  +  G  =  0 

(x-2)(x  —  3)(#-4)         =#3  —  9x2+26x— 24  =  0 
(#-2)(^-3)(#-4)(#-5)=#4-14#3+71#2-154#  +  120=0,&c. 
That  is,  the  product 

of  two  simple  equations  is  a  quadratic  equation  ; 

of  three  simple  equations,  is  a  cubic  equation  ; 

of  four  simple  equations,  is  a  biquadratic,  or  an  equa- 
tion of  the  fourth  degree,  &c.     (Art.  317.) 
27* 


318  GENERAL     PROPERTIES      OF     EQUATIONS. 

Or  a  cubic  equation  may  be  considered  as  the  product  of  a 
quadratic  and  a  simple  equation;  a  biquadratic,  as  the  product 
of  two  quadratic ;  or  of  a  cubic  and  a  simple  equation,  &c. 

In  each  case,  the  exponent  of  the  unknown  quantity,  in  the 

first  term,  is  equal  to  the  degree  of  the  equation ;  and,  in  the 

succeeding  terms,  it  decreases  regularly  by  1,  like  the  exponent 

of  the  leading  quantity  in  the  power  of  a  binomial.  (Art.  485.) 

In  a  quadratic  equation,  the  exponents  are         2,  1, 

In  a  cubic  equation,  3,  2,   1, 

In  a  biquadratic,  4,  3,  2,   1,  &c. 

The  number  of  terms,  is  greater  by  1,  than  the  degree  of 
the  equation,  or  the  number  of  simple  equations  from  which 
it  is  produced.  For  besides  the  terms  which  contain  the 
different  powers  of  the  unknown  quantity,  there  is  one  which 
consists  of  known  quantities  only.  The  equation  is  here  sup- 
posed to  be  complete.  But  if  there  are,  in  the  partial  pro- 
ducts, terms  which  balance  each  other,  these  may  disappear 
in  the  result.     (Art.  104.) 

•5.IO.    Each  of  the  values  of  the  unknown  quantity  is 
called  a  root  of  the  equation. 
Thus,  in  the  example  above, 

The  roots  of  the  quadratic  equation  are  3,  2, 

of  the  cubic  equation,  4,  3,  2, 

of  the  biquadratic,  5,  4,  3,  2. 

The  term  root  is  not  to  be  understood  in  the  same  sense 
here,  as  in  the  preceding  sections.  The  root  of  an  equation 
:s  not  a  quantity  which  multiplied  into  itself  will  produce  the 
equation.  It  is  one  of  the  values  of  the  unknown  quantity ; 
and  when  its  sign  is  changed  by  transposition,  it  is  a  term  in 
one  of  the  binomial  factors  which  enter  into  the  composition 
of  the  equation  of  which  it  is  a  root. 

The  value  of  the  unknown  letter  x,  in  the  equation,  is  a 
quantity  which  may  be  substituted  for  x,  without  affecting 
the  equality  of  the  members.  In  the  equations  which  we  are 
now  considering,  each  member  is  equal  to  0 ;  and  the  first  is 
the  product  of  several  factors.  This  product  will  continue 
to  be  equal  to  0,  as  long  as  any  one  of  its  factors  is  0.  (Art. 
106.)     If  then  in  the  equation 

(x-2)  X  (x-S)  X  (x-4)  X  (x-5)  =  0, 
we  substitute  2  for  x,  in  the  first  factor,  we  have 
0X(x-3)X(x-4)X(a;~5)=0. 


GENERAL     PROPERTIES     OF     EdUATIONS.        319 

So,  if  we  substitute  3  for  x,  in  the  second  factor,  or  4  in 
the  third,  or  5  in  the  fourth,  the  whole  product  will  still  be  0. 
This  will  also  be  the  case,  when  the  product  is  formed  by  an 
actual  multiplication  of  the  several  factors  into  each  other. 

Thus,  as     *3 -to2 +26^-24=0;   (Art.  535.) 
So  23-9X2*+26X2-24=0, 

And        |3-9X32  +26X3-24  =  0,  &c. 

Either  of  these  values  of  x,  therefore,  will  satisfy  the  con- 
ditions of  the  equation.  . 

We  have  thus  far  been  considering  higher  equations  as 
formed,  by  multiplication,  from  simple  equations.  But  the 
inquiry  may  arise,  whether  every  equation  of  a  higher  degree 
can  be  regarded  as  the  product  of  two  or  more  simple  equa- 
tions. It  is  proposed,  in  the  following  articles,  to  answer  this 
inquiry,  and  to  bring  into  view  a  number  of  the  most  impor- 
tant general  properties  of  equations. 

537.  An  equation  of  the  mth  degree  consists  of  xm,  the 
several  inferior  powers  of  x  with  their  co-efficients,  and  one 
term  in  which  x  is  not  contained.  l\  A,  B,  C, .  . . .  T,  be 
put  for  the  several  co-efficients,  and  U  for  the  last  term,  then 

xm+Axm- "  +Bxm~2  +  Cxm~ 3 +  Tx  -j-  {7=0, 

will  be  a  general  expression  for  an  equation  of  any  degree. 

Any  quantity  which,  substituted  for  xy  will  make  the  mem- 
bers equal,  is  called  a  root  of  the  equation.    (Arts.  335,  536.) 

If  (a)  is  a  root  of  the  general  equation  of  the  (m)th  degree, 
the  first  member  is  exactly  divisible  by  (x—a). 

For  by  substituting  a  for  x,  we  have 

am  +  Aam-1  +Bam-2  +  Cam-* +Ta+  [7=0. 

And  transposing  terms, 

U=  -am-Aam~l  -Bam~2  -  Cam~3 -  Ta. 

Substituting  this  value  for  U,  in  the  original  equation, 

xm  +  Axm- l  +Bxm-2  +Cxm- 3 +  Tx  | 

-am-Aam-l-Bam-2-Cam-* -Ta  ! 

Or,  uniting  the  corresponding  terms, 
(xm-<im)  +  A(xm-l-am-')  +  B(xm-2-am-2)-t- 
C(xm-*-am-3) +T(x-a)  =  0. 


=  0. 


320       GENERAL     PROPERTIES      OF      EQUATIONS. 

In  this  expression,  each  of  the  quantities  (xm  —  am), 
A(xm-1—am-l)}  &c.  is  divisible  by  x— a\  (Art.  130:)  there- 
fore the  whole  is  divisible  by  x  —  a. 

Conversely,  If  the  first  member  of  any  equation  be  divisi- 
ble by  x  —  a,  a  is  a  root  of  the  equation.  For  (Art.  114,)  this 
member  may  be  resolved  into  two  factors,  of  which  x  —  a  is 
one;  and  (Art.  106,)  it  must  itself  become  zero,  when  x  —  a, 
because  the  factor  x  —  a  becomes  zero.  The  equation  will 
therefore  be  satisfied,  by  giving  to  x  the  value  a;  and  a 
must  be  one  of  its  roots. 

Ex.  1.  Prove  that  2  is  a  root  of  the  equation 

x2-lx2  +  \2x— 4  =  0. 
This  is  to  be  done,  by  dividing  by  x— 2.     If  there  is  no 
remainder,  2  must  be  a  root  of  the  equation. 

Ex.  2.  Prove  that  3  is  not  a  root  of  the  equation 
x3+2x2  —  6x— 4  =  0. 

When  we  divide  here  by  x  —  3,  we  find  a  remainder;  which 
shows  that  3  is  not  a  root. 

Ex.  3.  Find  whether  —  1   is  a  root  of  the  equation 

x*—x3  —  5x2  +4^  —  3=0. 

Ex.  4.  Find  whether  1  is  a  root  of  the  equation 

x4-2x3-llx2-ar+15=0. 
Ex.  5.  Find  whether  —5  is  a  root  of  the  equation 

x5  +3x2  -6±x2  +x+23  =  0. 

t>38«  Every  equation  of  the  (m)th  degree  has  exactly 
(m)  roots. 

It  will  be  assumed  that  every  equation  has  at  least  one  root 
Let  a  be  a  root  of  the  equation 

xm+Axm-'+Bxm-2 +Tx+U=0. 

The  first  member  is  divisible  by  (x  —  a)  (Art.  537.)  If  we 
divide  by  this,  the  quotient  will  be  a  polynomial  of  the  degree 
(m—  1) ;  which  may  be  written  thus, 

xm~x  +A'xm-2  +B'xm~*  &c. 

If  we  make  this  equal  to  zero,  and  suppose  b  to  be  a  root 
of  the  equation  thus  formed,  the  first  member  will  be  divisi- 
ble by  x  —  b\  and  if  we  divide  by  this,  the  result  will  be  a 
polynomial  of  the  degree  (m  —  2).  By  proceeding  in  this 
way  till  we  have  divided   (m—  1)   times,  we  shall  obtain  a 


GENERAL     PROPERTIES      OF     ECIUATIONS.       321 

simple  equation,  having  only  one  root,  which  may  be  denoted 
by  /.  Hence  the  original  equation  has  m  roots,  a,  b,  c, . . .  /; 
and  the  first  member  of  it  is  composed  of  the  m  factors, 
x  —  a,  x—b,  x—c, . .  .  x—l. 

The  equation  can  have  no  other  root;  for  none  of  the 
factors  x  —  a,  x  —  b,  &c.  of  which  the  first  member  is  com- 
posed, can  be  zero,  unless  x  equals  one  of  the  quantities, 
a,  b,  &c. 

Ex.  1.  One  root  of  the  equation 

x3—4x2+2x+3=0 
is  3.     What  are  the  other  roots  ? 

If  we  divide  by  x—S,  we  obtain  the  equation  a;2 -#-1=0  ; 
which  must  contain  the  two  required  roots.     By  solving  this 

quadratic,  the  roots  will  be  found  to  be  — - —  • 

Ex.  2.  One  root  of  the  equation 

x3  —  5x*  +3.2+9=0 
is  3.     What  are  the  other  two?  Ans.  3  and  —  1. 

Ex.  3.  One  root  of  the  equation 

x3+5x2  —  3x— 7=0 
is  —1.     What  are  the  other  roots?  Ans.  —  2±v/ll. 

Ex.  4.  Two  roots  of  the  equation 

x4—  x3+2x2  —  14x  +  12  =  0 
are  1  and  2.    What  are  the  other  roots  ?     Ans.  -ldb\/-5. 
Ex.  5.  Two  roots  of  the  equation 

x*+x3  —  7x2—x+6=0 
are  1  and  —3.     What  are  the  other  roots? 
Ex.  6.  One  root  of  the  equation 

x4—  2x3+x2—  5x— 9=0 
is  —1.     Find  the  equation  containing  the  other  roots. 

•"539.  The  roots  of  an  equation  are  not  always  real. 
Some  or  even  all  of  them  may  be  imaginary.  In  the  fourth 
example  above,  two  of  the  roots  are  imaginary,  and  two  real. 

It  often  happens  that  the  roots  of  an  equation  are  not  all 
unequal.  Thus,  in  the  second  of  the  preceding  examples,  the 
roots  are  3,  3,  and  —1 ;  two  of  which  are  alike. 


322       GENERAL     PROPERTIES     OF     EQUATIONS. 

♦5  lO.  The  laws  by  which  the  co-efficients  of  an  equation 
are  governed,  may  be  seen,  from  the  following  view  of  the 
multiplication  of  the  factors 

x— a,  x— b,  x—c,  x—d, 
each  of  which  is  supposed  equal  to  0. 

The  several  co-efficients  of  the  same  power  of  x,  are 
placed  under  each  other. 

Thus,  —  ax—bx  is  written  ~?(#;  and  the  other  co- 
efficients in  the  same  manner. 

The  product,  then 
Of      (x-a)=0 
Into   (x— b)  =  0 

Is   £2     l  >  x+ab—0,  a  quadratic  equation. 
This  into   x— c=0 


This  into 


x— abc=0,  a  cubic  equation. 


abc^\ 

—      d  fxJ^a0C(^- 
-bcd) 


0,  a  biquadratic. 


5>4 1 .  By  attending  to  these  equations,  it  will  be  seen  that, 
In  the  first  term  of  each,  the  co-efficient  of  a:  is  1  ; 

In  the  second,  term,  the  co-efficient  is  the  sum  of  all  the 
roots  of  the  equation,  with  contrary  signs.  Thus  the  roots 
of  the  quadratic  equation  are  a  and  b,  anil  the  co-efficients, 
in  the  second  term,  are  —  «  and  —b. 

In  the  third  term,  the  co-efficient  of  xy  is  the  sum  of  all  the 
products  which  can  be  made,  by  multiplying  together  any  two 
of  the  roots.  Thus,  in  the  cubic  equation,  as  the  roots  are 
a,  b,  and  c,  the  co-efficients,  in  the  third  term,  are  ab,  ac,  be. 

In  the  fourth  term,  the  co-efficient  of  x  is  the  sum  of  all 
the  products  which  can  be  made,  by  multiplying  together  any 


GENERAL     PROPERTIES      OF     EQUATIONS.       323 

three  of  the  roots  after  their  signs  are  changed.  Thus  the 
roots  of  the  biquadratic  equation  are  a,  b,  c  and  d,  and  the 
co-efficients  in  the  fourth  term  are  —  abc,  —  abd,  —acd,  —bed. 
The, lasPlerm  is  the  product  formed  from  all  the  roots  of 
the  equation  after  the  signs  are  changed. 

In  the  cubic  equation,  it  is  —aX—bX—c=  —  abc. 

In  the  biquadratic,   —  aX—bX—cX—d=+abcd,  &c. 

542.  In  Art.  540,  the  roots  of  the  equations  are  all  rep- 
resented as  positive.  The  signs  are  changed  by  transposition, 
and  when  the  several  factors  are  multiplied  together,  the 
terms  in  the  product,  as  in  the  power  of  a  residual  quantity, 
(Art.  405,)  are  alternately  positive  and  negative.  But  if  the 
roots  are  all  negative,  they  become  positive  by  transposition, 
and  all  the  terms  in  the  product  must  be  positive.  Thus  if 
the  several  values  of  x  are  —a,  —b,  —c,  —d,  then 

x+a=0,  x+b=0,  x+c=0,  x  +  d=Q; 
and  by  multiplying  these  together,  we  shall  obtain  the  same 
equations  as  before,  except  that  the  signs  of  all  the  terms 
will  be  positive.     In  other  cases,  some  of  the  roots  may  be 
positive,  and  some  of  them  negative. 

Ex.  1.  Form  the  equation  whose  roots  are  1,  —1,  2,  —3. 

Ans.  x*+x3  —  7x2  —  x+6=0. 
This  result  may  be  obtained,  either  by  multiplying  together 
the  factors  x— 1,  x  +  1,  x—2,  x+S,  or  by  ascertaining  the 
co-efficients  according  to  the  law  in  Art.  541. 

f]x.  2.  Form  the  equation  whose  roots  are  1,  2,  —2,  3; 
both  by  multiplying  together  its  factors  x—  1,  x— 2,  &c.  and 
by  obtaining  the  co-efficients  from  the  law  in  Art.  541. 

Ans.  x*  —  4x3  —  x2+16x— 12=0. 
Ex.  3.  Find  the  equation  whose  roots  are  —1,  2,  —3. 

Ans.  x3+2x2  —  3a:— 6=0. 
Ex.  4.  Find  the  equation  whose  roots  are  1,  2,  2,  —2,  —3. 
Ans.  x5  —  llx3+6x2+28x  — 24  =  0. 

•113.  The  quotient  produced  by  dividing  the  original  equa- 
tion in  Art.  537,  by  x— a,  is  evidently  equal  to  the  aggregate 
of  the  particular  quotients  arising  from  the  division  of  the 
several  quantities  (xm-am),  A(xm-1 —am~*),  kc. 

The  quotient  of    (xm-am)  +  (x-a),  (Art.  130.)  is 
xm-l+axm-2+a2xn-3-\-a3xm-* . . .  +aw~1. 


324       GENERAL     PROPERTIES     OF     EQUATIONS. 

The  quotient  of   A(xm~l  —  am-")-ir(x—a)   is 
Axm-2+Aaxm-*+Aa2zm-* . . .  +^am"2. 
&c.  &c.        . 

Collecting  these  particular  quotients  together,  and  placing 
under  each  other  the  co-efficients  of  the  same  power  of  x, 
we  have  the  following  expression  for  the  quotient  of 

xm+Axm- ■  +Bxm~2  +  Cxm~ 3 +  Tx+  U 

divided  by  x— a. 


+a 

|»— !+«•      , 

)          +a3     ' 

\                    . 

. .+    am~* 

+A 

J                   +^« 

\xm~3+Aa'  1 

_ 

+Aam-* 

+B 

)          +Ba   \ 

+  C     . 

r  X 

+  Bam-* 
+  Cam~* 

And  if  b  is  a  root  of  the  equation,  the  quotient  from  dividing 
by  x~b,  is 

+4)  +Ab  [xm-3  +Ab2(    m_4        +7l6w-2 

+B    )  +Bb    (x  +Bbm~* 

H.  +C     J  +Cbm~* 


+T. 

And  if  c  is  another  root,  the  quotient  from  dividing  by  x-c,  is 

zm~l+c    I  Tm-2+c2    )  +c3     )  ...+    c7*"1 

+-4  J  +t!c  V^m-3-f^c2  f    m_4        +Acm~2 

+B    )  +Bc    (x  +I?c™-3 

III.  +C     J  +Ccm~* 

+T.' 

In  the  same  manner  may  be  found  the  quotients  produced 
by  introducing  successively  into  the  divisor  the  several  roots 
of  the  equation ;  which  are  equal  in  number  to  m. 

•511.  From  the  known  relations  between  the  roots  and 
the  co-efficients  of  equations,  as  stated  in  Art.  541,  Newton 
has  derived  a  method  of  determining  the  co-efficients,  from 
the  sum  of  the  roots,  the  sum  of  their  squares,  the  sum  of 
their  cubes,  &c,  though  the  roots  themselves  are  unknown ; 
and  on  the  other  hand  of  determining  from  the  co-efficients, 
the  sum  of  the  roots,  the  sum  of  their  squares,  the  sum  of 
their  cubes,  &c.  For  this  purpose,  the  following  plan  of  no- 
tation is  adopted.     Sx  is  put  for  the  sum  of  the  roots,  S2  for 


GENERAL     PROPERTIES     OF     ECIUATIONS.       325 

the  sum  of  their  squares,  S3  for  the  sum  of  their  cubes,  &c. 
If  the  roots  are  a,  b,  c,  d, . . .  /,  then 

Sl=a+b+c+d...+l 

S2  =a2  +  b2  +c2  +d%. .  +Z3 

Sm=am+bm  +  cm+dm. .  .  +  lm  S 
Sm-,  =am~'  +bm~l  +C7""1  +dw-1 . . .  +/m"1 
&c.  &c. 

By  means  of  this  notation,  we  obtain  the  following  expres- 
sion for  the  sum  of  all  the  quotients  marked  I,  II,  III,  &c. 
(Art.  543,)  and  continued  till  iheir  number  is  equal  to  m. 

wta-^+S,    )     „.,+    S2)  +    Si)  ...+    Sm-l 

+mA\x       +ASX  \x"-*+ASa  (    _4       +ASm-M 

+mB   )  +BSAX  +BSm-9 

Y.  +mC  )  +C&.-4 

fmT. ' 

In  the  original  equation, 

a;ro+^a;w-1  +  JRcw-a  +  Ckw-3 . .  .  +  Tx+  [7=0, 
the  co-efficients,  A,  B,  C,  &c.  have  determinate  relations  to 
the  sum  and  products  of  the  roots,  a,  b,  c,  &c.  (Art.  541.) 
But  the  quotient  marked  I,  (Art.  543,)  produced  by  dividing 
by  x  —  a,  is  the  first  member  of  an  equation  of  the  next  infe- 
rior degree,  from  which  the  root  a  is  excluded,  since  it  is 
the  product  of  x—b,  x—c,. .  .x—l,  (Art.  538.)  So  b  is  ex- 
cluded from  the  quotient  II,  c  from  the  quotient  III,  &c.  In 
the  expression  above  marked  Y,  which  is  the  sum  of  m  quo- 
tients, the  co-efficient  of  x  in  the  second  term  is  Si  +mA. 
But  A,  which  is  the  co-efficient  of  x  in  the  second  term  of 
the  original  equation,  is  equal  to  the  sum  of  the  roots  a,  b,  c, 
&c.  with  contrary  signs;    (Art.  541,)  that  is,  Si  =  —A. 

Therefore, 

Sl+mA=(m—l)A. 

In  the  third  term  of  the  original  equation,  B  the  co-efficient 
of  x,  is  equal  to  the  sum  of  all  the  products  which  can  be 
made  by  multiplying  together  any  two  of  the  roots.  (Art.  541.) 
But  each  of  these  products  will  be  excluded  from  two  of  the 
quotients,  I,  II,  III,  &c.  For  instance,  ab  will  not  be  found 
in  the  first,  from  which  a  is  excluded,  nor  in  the  second, 
from  which  b  is  excluded.     Therefore  in  the  expression  Y, 

2ft 


326         GENERAL     PROPERTIES      OF     EQUATIONS. 

the  co-efficient  of  x  in  the  third  term  is  equal  to  mB—2ab— 

2ac-2ad,  &c.     But  —  2ab,  -2ac,  -2ad,  &c.  =-2£.     So  that 

S2+ASl+mB=(m-2)B. 

In  the  fourth  term  of  the  original  equation,  C  the  co-effi- 
cient of  a?,  is  equal  to  the  sum  of  all  the  products  which  can 
be  made  by  multiplying  together  any  three  of  the  roots,  after 
their  signs  are  changed.  But  each  of  these  products  will  be 
excluded  from  three  of  the  quotients,  I,  II,  III,  &c.  So  that, 
in  the  expression  F,  the  co-efficient  of  x  in  the  fourth  term, 
is  equal  to  mC—Sabc—Sabd,  &c.     That  is, 

S3+AS2+BSi+mC=(m-3)C. 

In  the  same  manner,  the  values  of  the  co-efficients  of  x  in 
succeeding  terms  may  be  found  ;  the  number  of  the  co-effi- 
cients being  one  less  than  the  number  of  roots  in  the  equation. 

Collecting  these  results,  we  have 

Si-\-mA=(m-\)A, 
S2+ASl+mB=(m-2)B, 
S3+AS2+B$1+mC=(m-3)C, 
St+AS3+BS2  +  CS1+mD=(m-4)Di 
&c.  <kc. 

Transposing  and  uniting  terms, 
I  £,+,4=0, 

S2+ASl+2B=0, 
S3+AS2+BSl+3C=0, 
S*  +AS3+BS2  +  CSx  +4D=0, 
&c.  &c. 

Substituting  for  Si$  S2i  S3,  &c.  their  values,  and  reducing, 
H  S,  =  -Af 

S2=     A*-2B, 
S3  =  -A*+3AB-3C, 
S<=    A*-4A2B+4AC+2B*-4D, 
&c.  &c. 

We  have  here  obtained  symmetrical  expressions  for  the 
sum  of  the  roots  of  an  equation,  the  sum  of  their  squares, 
the  sum  of  their  cubes,  &c.  in  terms  of  the  co-efficients. 
By  transposing  the  terms  in  the  expressions  marked  I,  we 
have  the  following  values  of  .A,  Bt  C,  &c. 


GENERAL     PROPERTIES     OF     EQUATIONS.       327 
III.  A=*r-Sj 

C=t-±(BSi+A$*+Ss) 

D=-\(CSl+BS2+AS3+S,) 
&c.  &c. 

By  which  the  co-efficients  of  an  equation  may  be  found  from 
the  sum  of  its  roots,  the  sum  of  their  squares,  the  sum  of 
their  cubes,  &c. 

Ex.  1.  Required  the  sum  of  the  roots,  the  sum  of  their 
squares,  and  the  sum  of  their  cubes,  in  the  equation 

x4~10x-3+35x2-50x-24  =  0. 
Here    .4= -10.  5=35.  C=-50. 

Therefore  Si  =  10 

£2  =  102-(2X35)=30 

S9  =  10*  +  (3X-  10X35)-  (3X-50)  =  100. 

2.  Required  the  terms  of  the  biquadratic  equation  in  which 
#,  =  1,  $2=39,  #3  =  —  89,  and  the  product  of  all  the  roots 
after  their  signs  are  changed  is   —30. 

Ans.  xA—  x3  —  19x2+49:c— 30=0. 

•>  1*5.  There  is  occasion  in  some  of  the  following  inves- 
tigations to  substitute  the  sum  of  two  quantities  for  a  single 
quantity  in  a  given  polynomial.  When  the  terms  of  the 
result  are  arranged  according  to  the  powers  of  one  of  the 
new  quantities,  the  co-efficients  of  this  quantity  are  found  to 
follow  a  remarkable  law :  and  may  be  readily  derived  from 
each  other. 

In  the  polynomial 

xm+Axm-1+Bxm-2  &c. 
let  r+y  be  put  for  x ;  it  will  then  be 

{r+y)m  +  A(r+y)m-l-\-B(r+y)m~2   &c. 

And  if  we  develope  the  powers  of  r+y  by  the  binomial 
theorem,  and  arrange  the  resulting  terms  according  to  the 
powers  of  y}  we  have 

rm+Arm-l+Brm-2  &c. 
+  (mrm-l+(m-l)Arm-*  +  (m-2)Brm-3  &c.)  y 
'  (m(m-\)  m  _    (m-l)(m-2)  .            (m-2)(m-3)  „         .     \ 
+  \      2  + 2 Arm~3+ ^ LBrm-*&c.)?j* 


328       GENERAL     PROPERTIES      OF     EClUATIONS* 

/m(m-l)(m-2)  (m-l)(m-2){m-3) 

™  2.3  ^  2.3 

(?/2-2)(ra— 3)(?rc  — 4)„  c      \ 

* 0  Q  A -J3rw-5   &c.)y3  +  &c. 

For  the  sake  of  brevity,  we  will  put 
R=j^-\-Arm'1+Brm-2  &c. 
JR'=mrw»-i+(7/2-l)7lrm-2+(m-2)J5rm-3  &c. 
R"=m(m-l)rm-2  -{-(m-l)(m-2)Arm-3  +  (m-2)(m-3)Brm-^  &c. 
7J///=w(;/i-l)(7?2-2)rm-3  +  (m-l)(m-2)(w-3)^lr7B-4 
+  (m-2)(m-3)(m-4)£rm-5  &c. 
&c.  &c. 

The  preceding  polynomial  then  becomes 

R"        R" 

R+R'y+-^-y2  +—y*  &c. 

And  it  may  be  seen  that 

R  is  derived  from  the  given  polynomial,  by  substituting  r  for  x; 

R  is  derived  from  R  by  multiplying  each  term  by  the  expo- 
nent of  r  in  that  term,  and  diminishing  the  exponent  by 
unity ; 

And  each  of  the  polynomials  R",  R'",  &c.  is  derived  from 

the  preceding  one,  in  the  same  manner  as  jR'  from  R. 
R'  is  called  the  first  derived  polynomial  of  Ry 
R"  the  second  derived  polynomial  of  R, 

or  the  first  derived  polynomial  of  R't 

and  so  on. 

Ex.  1.  Find  the  successive  derived  polynomials  of 
r3  —  ra+2r+6.  Ans.  3r2— 2r+2,  6r— 2,  6. 

Ex.  2.  Find  the  derived  polynomials  of   %4-f2z2  — 6z— 4. 

Ans.  4z3"+4z  — 6,   12z3+4,  24z,  24. 
Ex.  3.  Find  the  derived  polynomials  of  x5— 3^4— 3^2+5. 
Ans.  5x*-l2x3-Gx,  20x3-36x2-G,  60:r2-72x,  120^-72,  120. 
Ex.  4.  What  are  the  derived  polynomials  of 
.t4-x3+2x2-3x+4? 

546.  To  transform  an  equation  into  another  whose  roots 
shall  he  greater  or  less  than  those  of  the  former,  by  any  given 
quantity. 


GENERAL     PROPERTIES      OF     EQUATIONS.       329 

Let  the  given  equation  be 

xm+Axm"l+Bxm'a  &c.  =0; 
which  it  is  required  to  transform  into  another,  whose  roots 
are  less  than  its  own,  by  r.     Put  y—x—i\     Then  x=r+y; 
and  by  substitution,  the  given  equation  will  become 


,2  7/3 


R+R'y+R"£-+R'"g-=  &c.  =0; 

the  polynomials  R,  R',  R",  &c.  being  determined  by  the  rule  in 
the  preceding  article.  Since  y=x—r,  the  values  of  y  which 
will  satisfy  the  last  equation,  are  less  by  r  than  those  of  x  in 
the  former  equation  ;  that  is,  the  roots  of  the  new  equation, 
in  which  y  is  the  unknown  quantity,  are  less  by  r  than  those 
of  the  original  equation  in  which  the  unknown  quantity  is  x. 
To  render  the  roots  of  the  new  equation  greater  by  r  than 
those  of  the  original  equation,  we  have  only  to  change  the 
sign  of  r,  putting  y= x-fr. 

Ex.  1.  Transform  the  equation 

x3-lx2-  lis +42=0 
into  another,  whose  roots  are  less  than  those  of  the  first  by  r. 
Put  y=x—r,   then  by  Art.  545, 

R    =r3_7r2_llr+42 

R'   =  3r2-14r-ll 

#"=67—14 

JR"'=6. 

Hence, 

(r3-7r2-llr+42)  +  (3r2-14r-ll)?/+-^— y3  +  —-y*=0, 

or  writing  this  in  the  usual  way, 
y3  +  (3r-7)?/2+(3r2-14r-ll)z/-fr3~7ra-llr-f42=0. 

Ex.  2.  Transform 

x3+x2-Ux+\2=0, 
into  an  equation  whose  roots  are  greater  than  those  of  the 
former  by  r. 

Put  y—x+r.     The  derived  polynomials,  if  r  had  the  con- 
trary sign,  would  be 

R       =,r3+r2_14r+12 

R   =3r2+2r-14 
R»  =6r+2 
R'"=G. 

28* 


330       GENERAL     PROPERTIES      OF     EdUATIONS. 

Then  by  changing  the  signs  of  the  odd  powers  of  each,  and 
remembering  that  R"  is  to  be  divided  by  2,  and  R"  by  2.3, 
or  6,  we  shall  obtain  the  co-efficients  of  y  in  the  required 
equation  ;  which  accordingly  is 

y3_(3r— l)y2  +  (3r2— 2r— 14)y— r3+r2-f  14r+12  =  0. 

Ex.  3.  Find  the  equation  whose  roots  are  greater  by  2 
than  those  of  the  equation 

x4  — 2x3  -f-5z2  +4^—8=0. 

Ans.  y4~10?/3+4l2/a-72y+36=0. 

Ex.  4.  Find  the  equation  whose  roots  are  less  by  1  than 
those  of  the  equation 

x5+3x3-7x2+x+l2=0. 
Ans.   ?/5+5?/4  +  132/3  +  122/2+y+10  =  0. 

Ex.  5.  What  is  the  equation  whose  roots  are  greater  by  1 
than  those  of  the  equation 

a:4—  5x2  —  6x— 2  =  0? 

Ans.  y*  —  4y3+y2=0. 

One  root  of  the  last  equation  is  obviously  zero.  And  if 
we  divide  by  y— 0,  that  is  by  y,  the  result  is  y3-4y2  +?/=0; 
which  also  has  zero  for  one  of  its  roots.  Dividing  again  by 
y,  we  obtain  the  quadratic  equation  y2  —  4y-fl=0 ;  whose 
roots  are  2d=x/3.  From  this  we  see  that  the  roots  of  the 
original  equation  must  be  —1,   -—1,  l+v/3,  1—^/3. 

547.  There  is  another  method  of  transforming  equations, 
so  as  to  increase  or  diminish  their  roots,  which  is  often  more 
convenient,  than  the  preceding.  In  employing  it,  we  have 
occasion  to  perform  a  number  of  successive  divisions.  They 
are  all,  however,  simple  and  of  a  similar  kind.  Before  pro- 
ceeding to  explain  the  method,  we  will  show  how  these  divis- 
ions are  most,  conveniently  performed.  In  each  of  them,  the 
divisor  is  a  binomial  of  the  form  x  —  a,  and  the  dividend  a 
polynomial,  as  Ax  +  B,  Ax2+Bx+C,  Ax3  +  Bx2 +  Cx  +  D, 
or  one  of  some  higher  degree  with  respect  to  x.  Let  the 
proposed  dividend  be 

Ax*+Bx3  +  Cx2  +  Dx+E. 
If  we  divide   this  by  x— a,    the   first   term  of  the  quotient 
will   be   Ax3.     Let   the  whole  quotient   be   represented    by 
Ax3+B'x2  +  C'x+D\  and  the  remainder  by  E'.     Then,  as 


GENERAL     PROPERTIES      OF     EQUATIONS.       331 

the  product  of  the  divisor  and  quotient,  together  with  the 
remainder,  is  equal  to  the  dividend,  we  have  the  equation 
Ax*+Bx*+Cx*+Dx+E=(x-a)(Ax*+B'x*+C'x+D')+E' 
=Ax*  +  (B'-aA)x3  +  (C'-aB')x2  +  (Df-aC,)x+E^aDJ 

Hence,  B'-aA=B    and  B'=B+aA 

C'-aB'=C  C'=C+aB' 

D'-aC'=D  D=D+aC' 

E'-aD'=E  E=E+aD' 

By  these  last  formulas,  it  is  easy  to  compute  successively 
the  unknown  co-efficients  of  the  quotient,  namely,  B\  C,  D\ 
and  the  remainder  E'. 

The  same  method  may  be  pursued,  whatever  be  the  degree 
of  the  dividend.     Hence, 

To  divide  a  polynomial  of  the  form 

Axm  +  Bxm- \  +  Cxm'2 . . .  +  Tx+  U, 
by  the  binomial    x—a, 
Represent  the  quotient  by 

Axm-'+B'xm-2  +  C,xm'3J^[p^1R' 
and  the  remainder  by  U'.  «TM»  t  rr 

Then  find  the  co-efficient  vT/>  <    ^ 

B'     by  the  equation         ^'  =  *^^vJpf;T<  $\^ 
C     by  the  equation         C'—C+c 
&c.  <fec. 

the  final  equation  being  U'=  U+aT. 

To  divide  by  x+a  instead  of  x—a,  we  have  only  to 
change  the  sign  of  a  in  the  preceding  formulas,  making 

B'  =  B-aA 
C'=C-2Bf 

&c. 

Ex.  1.  Divide   2x*+lx*-8x2-\\x  +  \8   by  x-3. 

264 

Ans.  2^3  +  13x2+31:r+82H -. 

x— 3 

The  process  for  obtaining  this,  according  to  the  rule,  is  as 
follows. 


332 


GENERAL 

PROPERTIES 

OP 

E  aUATIO  NS 

A 

B 

G 

D 

E 

2 

7 

-  8     - 

-11 

18 

6 

39 

93 

246 

13 

31 

82 

264 

B' 

O 

D' 

E' 

Here  2  is  multiplied  by  3  and  the  product  added  to  7 ;  the 
sum  13  or  B'  is  multiplied  by  3  and  the  product  added  to 
—8;  and  so  on. 

Ex.  2.   Divide  x5  —  4x4  +9x3  +  12x2  —  lOx— 2  by  x— 2. 
The  required  operation  is  as  follows. . 

1-4  9         12     -10     -  2 

2       -4  H?         44         68 

^      ""1  22  34         66 

Ans.  x4-2x3+5x2+22x+34  + 


x-2 

The  work,  here  and  in  other  cases,  may  be  shortened  by 
omitting  to  write  the  second  line  of  numbers.  Thus,  in  the 
preceding  case,  when  1  is  multiplied  by  2  and  the  product 
added  to  —4,  we  may  omit  to  write  the  product  2,  and  set 
down  only  the  sum  —2.  In  like  manner  we  may  omit  the 
product  —4,  and  write  only  the  sum  5;  and  so  on.  The  whole 
operation  will  then  be  presented  in  this  abbreviated  form, 
1-4  9  12  -10  -  2 
-2         5         22  34  66 

Ex.  3.  Divide   2x4  —  5x2  —  14x— 20   by  x—  1. 

37 

Ans.  2x3+2x2-3x-17 -• 

x—  1 

When,  as  in  this  example,  any  term  of  the  dividend  is 
wanting,  the  place  of  its  co-efficient  must  be  supplied  with 
zero. 

Ex.  4.   Divide    5x3-5x2  +  10x-12   by  x+2. 

92 

Ans.   5x2  — 15x+40 — :• 

x+2 

Ex.  5.  Divide   x5  —  5x4—  6x3—  x+1   by  x+1. 

2 

Ans.    x4—  6x3  —  H —* 

x+1 

548.  We  will  now  explain  the  method  of  transforming 
equations,  by  successive  divisions,  in  order  to  increase  or 
diminish  their  roots. 


GENERAL     PROPERTIES     OF     EQUATIONS,       333 

Let  it  be  required  to  transform 

Ax*  +Bx*  +  Cx2  +  Dx+ jE=0 
into  another  equation  whose  roots  are  less  than  those  of  the 
former  by  a.     If  we  take  y=x—a,  that  is,  x— y+a,  and  sub- 
stitute this  value  for  x  in  the  above  equation,  we  shall  obtain 
the  equation  required.     This  equation  will  be  of  the  form 

Ay*+B'y3  +  C'y2+D'y+E'=0. 
In  order  to  determine  the  co-efficients  Bf,   C,  D'  and  E' 
more  conveniently  than  by  the  substitution  just  mentioned, 
we  observe  that  if  x—a  be  put  for  y,  the  last  equation  will 
be  converted  into  the  original  equation ;  that  is, 
Ax"  +Bx3  +  Cx2  +Dx+E 

=A(x-a)*+B'(x-a)3  +  C,(x-a)2+D'(x-a)+E' 

must  be  an  identical  equation  (Art.  173.)  Now  if  we  divide 
the  second  member  by  x— a,  the  remainder  will  be  E',  and 
the  quotient 

A(x-a) 3  +  B'(x-a)2  -f  C'(x-a) +D' ; 
and  if  we  divide  the  latter  by  x— <z,  the  remainder  will  be 
D',  and  so  on ;  the  unknown  quantities  E\  Df,  C,  B'  being 
the  remainders  of  the  successive  divisions.  Then  if  the 
first  member  be  divided  in  the  same  way  by  x— a,  the  re- 
mainders of  the  successive  divisions  must  be  the  same, 
namely,  E\  D\  C,  B' ;  and  thus  these  quantities  may  be 
easily  determined. 

In  a  similar  manner,  the  co-efficients  of  the  required 
equation  may  be  found,  whatever  be  the  degree  of  the  given 
equation.     Hence, 

To  transform  an  equation  into  another  whose  roots  are 
less  than  those  of  the  former  by  a  given  quantity  a>  we  have, 
in  addition  to  the  method  in  Art.  546,  the  following  rule. 

Divide  the  first  member  of  the  equation  by  x—a,  (x  being 
the  unknown  quantity ;)  reserve  the  remainder  and  divide  the 
quotient  by  x—a;  reserve  the  new  remainder  and  divide  the 
new  quotient  by  x—a;  and  so  on.  The  first  remainder  will 
be  the  last  term  of  the  required  equation ;  the  second  remain- 
der will  be  the  co-efficient  of  y,  the  unknown  quantity ;  the 
third  remainder  will  be  the  co-efficient  of  y2  ;  &c. 

If  the  roots  of  the  given  equation  are  to  be  increased  by 
a,  we  must  divide  by  x-\-a,  instead  of  x—a. 

The  divisions  are  to  be  made  by  the  rule  in  the  preceding 
article. 


3 

-1 

-4 

-2 

2 

2 

1 

1 

-2 

6 

14 

7 

15 

14 

21 

334      GENERAL     PROPERTIES     OF     ECtUATIONS. 

Ex.  1.  Find  the  equation  whose  roots  are  less  by  2  than 
those  of  the  equation 

2x*—5x3+3x2—  x— 4=0. 

The  required  divisions  may  be  performed  as  follows, 

2         -5 

4 

-1 

4 

3 

4 

7 
4 

11 

The  divisor  here  being  x— 2,  the  co-efficients  of  the  first 
quotient,  obtained  as  in  Art.  547,  are  2,  —1,  1,  1,  and  the 
remainder  —2.  All  except  the  first  of  these  are  in  the  third 
line  of  numbers  above  ;  the  first  co-efficient  2,  being  the  same 
as  that  of  the  original  equation,  is  at  the  beginning  of  the 
first  line.  The  remainder  —2  being  reserved,  and  the  quo- 
tient, (whose  co-efficients  have  been  found  to  be  2,  —1,  1,  1,) 
being  divided  by  x—2,  (Art.  547,)  the  co-efficients  of  the 
new  equation  are  found  to  be  2,  3,  7,  and  the  remainder  15. 
Another  division  gives  2  and  7  for  the  co-efficients  of  the 
quotient,  and  21  for  the  remainder.  And  the  division  ends  by 
giving  the  quotient  2  and  the  remainder  11.  The  required 
equation  therefore  is 

2y4-Hl*/3+21y2-r-15?/-2=0. 

Ex.  2.  Find  the  equation  whose  roots  are  less  by  the  deci- 
mal .  1  than  those  of  the  equation 

2y4-f  ll2/3-f21y2+15?/-2  =  0. 

The  required  operation  is  as  follows. 


11 

21 

15 

-2 

.2 

1.12 

2.212 

1.7212 

11.2 

22.12 

17.212 

-  .2788 

.2 

1.14 

2.326 

11.4 

23.26 

19.538 

.2 

1.16 

11.6 

24.42 

.2 

11.8 


GENERAL     PROPERTIES 

OF     E  QUATIONS.       3 

re  briefly,  (Art.  547> 

,  Ex.  2,) 

2         11 

21 

15              -2 

11.2 

22.12 

17.212     -    .2788 

11.4 

23.26 

19.538 

11.6 

24.42 

11.8 

Ans.  2z4  +  11.8z3 

+24.42* 

2  +  19.538z-.2788=0. 

335 


Ex.  3.    Increase  by  3  the  roots  of  the  equation 

x5+2x*+3x,J—  4x-5=0. 
Here  the  co-efficients  are  1,  0,  2,  3,  —4,  —5.     We  then 
proceed  as  follows, 

1  0         2 

-3         9 


-3       11 
-3       18 


-6       29 

-3       27 


3 
-33 

-4 

90 

-5 
-258 

-30 

-87 

86 
351 

-263 

-117 
-168 

437 

-9       56     -285 
-3       36 


-12       92 
-  3 


Or 


-15 


2 

3 

-4 

-5 

11 

-30 

86 

-263 

29 

-117 

437 

56 

-285 

92 

1  0 

-3 

-6 

^  -9 

-12 

-15 

Ans.   t/5-152/4+92y3-285?/2+437*/-263=0. 

Hx.  4.  Diminish  by  .02  the  roots  of  the  equation 

2x4—  x3  —  10x+12=0. 
Ans.  2y4-.84y3-.0552*/2-10. 001 136y+ 11 .79999232=0. 
Ex.  5.  Diminish  by  .03  the  roots  of  the  equation 

4x3-.36.r2  +  .  08*4-2.15=0. 

Ans.   4y3-K0692y +2. 152184=0. 

5>  ID.   It  is  sometimes  required  to  transform  an  equation 
into  another  whose  second  term  is  wanting. 


336       GENERAL     PROPERTIES     OF     EQUATIONS. 

Let  the  given  equation  be 

xm+Axm-l+Bxm-2  &c.  =0. 

If  in  this  we  substitute  y+r  for  x,  and  develope  the 
powers  of  y+r  by  the  binomial  theorem,  we  see  that  no 
power  of  y  as  h'gh  as  ym"[  can  result,  except  from  the  first 
two  terms,  namely,  xm  and  Axm~l,  that  is,  (y+r)m  and 
A{yJrr)m~x.  The  first  of  these  developed  is  ym+mrym~A  &c. 
The  second  is  Ay™'1  +  (m  —  \)Arym'2  &c.  Taking  their 
sum,  we  find  the  term  containing  ym~l  to  be  {mr-\-  A)ym~  l. 
In  order  to  make  this  disappear,  so  that  the  new  equation, 
having  y  for  its  unknown  quantity,  may  be  of  the  form 
tj'i  +  B'yn-z  +  C'y™-*  &c.  =0, 

we  must  suppose  that  mr+A=0;  in  which  case  r= ; 

and  y+r,  that  is  x,  —y • 

y  *      m 

Hence,  to  make  the  second  term  of  an  equation  disappear, 
Substitute  for  the  unknown  quantity,  a  new  unknown  quan- 
tity,  together  with   the  co-efficient  of  the  second  term,  taken 
with  a  contrary  sign,  and  divided  by  the  number  answering 
to  the  degree  of  the  equation. 

Ex.  1.  Transform  x3  —  6x2-\-7=0  into  an  equation  whose 
second  term  is  wranting.  Ans.  y3  —  \2y—9=0. 

Here  we  substitute  for  x  the  new  unknown  quantity  y, 
together  with  +2,  which  is  a  third  of  the  co-efficient  -f6, 
taken  with  the  contrary  sign. 

Ex.  2.  Transform  x3-\-Sx2  —  Sx— 1=0,  into  an  equation 
whose  second  term  is  wanting.  Ans.  y3  —  6y+4=0. 

Ex.  3.  Transform  x*  —  Sx3  —  5x+12=0  into  an  equation 
whose  second  term  is  wanting. 

Ans.  y*  —  2iy2—  69y— 46  =  0. 

Ex.  4.  Transform  x5  +20.r4-f2.r2  —  Sx  —  24=0  into  an 
equation  whose  second  term  is  wanting. 

•>*>!).    If  the  signs  of  the  alternate  terms  of  an  equation 
be  changed,  the  signs  of  all  the  j*oots  will  be  changed. 
Let  a  be  one  of  the  roots  of  the  equation 

xm+ Axm-*+  Bxm-2  +  Cxm~3  &c.  =0; 
then  —a  is  a  root  of  the  equation 

xm— Axm~ l  +Bxm-*  -  Cxm~3  &c.  =0; 


GENERAL     PROPERTIES     OF     EQUATIONS.       337 

that  is,  —  a,  when  put  for  x  will  reduce  the  first  member  to 
zero.  For  when  —a  is  substituted  for  x,  the  first  member 
becomes 

am+Aam-l+Bam-2  +  Cam-*  &c. 
if  m  is  an  even  number ;  and  it  becomes 

-am-Aam-*-Bam-2-Cam-3  &c. 
that  is,         -(am+Aam-1  +  Bam~2  +  Cam~3  &c), 

if  m  is  odd.     The  value  in  either  case  must  be  zero.     For 
since  a  is  a  root  of  the  equation  first  proposed, 
am+Aam-'+Bam-2  +  Cam-3  &c.  =0. 

In  this  demonstration,  it  is  implied  that  the  equation  is 
complete ;  or  at  least  that  its  terms  contain  alternately  odd 
and  even  powers  of  x.  When  this  is  not  the  case,  we  should 
supply  the  place  of  every  term  that  is  wanting,  with  zero ; 
so  as  to  render  the  equation  complete  in  form,  before  apply- 
ing to  it  the  preceding  proposition. 

Ex.  1.  The  roots  of  the  equation 

x3+4x2+x— 6=0, 
are  1,  2,  —3.     What  are  those  of  the  equation 
x3—  4x2  -fx-}-6=0? 
Ex.  2.  The  roots  of  the  equation 

x4-4x3-7x*  +34#-24=0, 
are  1,  2,  —3,  4.     What  are  those  of  the  equation 
x*  -\-4x3  -7x3  -34cc-24=0  ? 
Ex.  3.  The  roots  of  the  equation 

<c4+4:c3-f-3x3—  4x— 4=0, 
are  1,  —1,  —2,  —2.     What  are  those  of  the  equation 
x*  -4x*  +3x2  +4a-4=0  ? 
Ex.  4.   The  roots  of  the  equation 
x3— 7;r-f6=0 
are  1,  2,  and  —3.     What  is  the  equation  whose  roots  are 
-1  -2  and  3? 

t>ol.    Through  the  remainder  of  this  section,  with  the 

exception  of  Arts.  5ft  1  563,   it  is  to  be  understood  that 

the  co-efficients  A,  B,  C. . .  T,  U,  occurring  in  equations  or 

polynomials,  are  all  real  quantities. 

29 


338       GENERAL     PROPERTIES     OF     EClUATIONS. 

5«>2«   In  any  polynomial  of  the  form 

xm+Azm- ]  +  Bxm~2 . . .  +  Tx+  U, 
a  value  can  be  given  to   (x)   sufficiently  large  to  render  the 
first  term  greater  than  the  sum  of  all  the  rest. 

To  take  the  most  unfavorable  case,  suppose  A,  B, ...  U  to 
be  all  positive,  or  all  negative.  If  L  be  put  for  the  largest  of 
these  quantities,  the  sum  of  all  the  terms  after  the  first  will 
be  less  than 

Lxm~ ■ +Lxm~2 . . .  +Lx+L, 

that  is,   L(xm-*+xm-2...+x  +  l)  or  (Art.  452,)  Lx*—^-- 
Now  in  order  that  this  quantity  be  less  than  xM,  it  is  suffi- 
cient that  x  be  equal  to  £+1.     For  ->  will  then  be  equal 

x      1 

xm—  1 
to  unity,  and  £x     _  -  equal  to  xm—\  ;  which  is  less  than 

xm.     Hence  the  sum  of  the  terms 

Axm~*  +Bxm~l . . .  +  Tx+  U 
must  also  be  less  than  xm. 

Any  value  of  x  greater  than  £+1,  will  likewise  render  the 
first  term  of  the  given  polynomial  greater  than  the  sum  of 
all  the  rest. 

Ex.  1.  What  value  of  x  will  render  the  first  term  of  the 
polynomial 

x*  +9x3  +7x2  —  lOx— 8 
greater  than  the  sum  of  all  the  rest  ? 

Here  the  largest  co-efficient  is  10.  Increasing  this  by  1, 
we  have  11  for  the  required  value  of  x. 

Ex.  2.  What  value  of  x  will  make  the  first  term  of  the 
polynomial  »»_ii«._io*_9 

greater  than  the  sum  of  the  other  terms  ? 

Ex.  3.  What  value  of  x  will  make  the  first  term  of  the 
polynomial  x<+lx>+6x>-lx+5 

greater  than  the  sum  of  the  others  ? 

The  above  theorem  will  be  employed  in  the  proof  of  some  of 
the  following  propositions  concerning  the  roots  of  equations. 


GENERAL     PROPERTIES     OP     EdUATIONS.       339 

5 *ilt.  If  two  numbers,  when  substituted  for  the  unknown 
quantity  in  the  first  member  of  an  equation,  give  results  of 
different  signs,  the  equation  has  at  least  one  root  comprised 
between  those  numbers. 

Let  a  certain  number  p,  when  put  for  x  in  the  equation 

xm+Axm-1-{-Bxm-2  &c.  =0, 

make  the  first  member  positive ;  and  another  number  q  ren- 
der it  negative :  the  equation  must  have  at  least  one  root 
between  p  and  q. 

By  supposing  x  to  vary  in  value  uninterruptedly,  the  first 
member  of  the  equation  will  be  made  to  vary  uninterruptedly. 
But  x  in  varying  from  p  to  q,  makes  the  first  member  of  the 
equation  pass  from  a  positive  to  a  negative  value.  Now  a 
quantity,  varying  uninterruptedly,  can  not  change  its  value 
from  positive  to  negative,  without  first  becoming  zero.  There 
must  therefore  be  some  value  of  x  between  p  and  q,  which 
satisfies  the  proposed  equation,  making  its  first  member  equal 
to  zero ;  that  is,  there  must  be  at  least  one  root  of  the  equa- 
tion between  the  numbers  p  and  q. 

If  the  numbers  p  and  q  differ  only  by  unity,  the  smaller 
number  is  the  integral  part  of  the  root  comprized  between 
the  two. 

Ex.  1.  What  is  the  integral  part  of  a  root  of  the  equation 

x3— lx2+3x— 4=0? 

It  will  be  found,  by  trial,  that  the  contiguous  numbers  6 
and  7,  when  substituted  for  x  in  the  first  member  of  the 
equation  give  results  with  contrary  signs.  Hence  the  equa- 
tion has  a  root  comprized  between  6  and  7.  The  integral 
part  is  therefore  6. 

Ex.  2.  Find  the  integral  parts  of  two  roots  of  the  equation 

x*  —  l2x3-20x2-36=0. 

Ans.   13  and  —  2. 

Ex.  3.  Find  the  integral  parts  of  two  roots  of  the  equation 
x*—  4x3+5x2—  2x— 6=0. 

Ans.  2  and  0. 

Ex.  4.  What  is  the  integral  part  of  a  root  of  the  equation 
#3-a;2+7=0? 


340       GENERAL     PROPERTIES     OF     EQUATIONS. 

•Tol.  Every  equation  of  an  odd  degree  has  at  least  one 
real  root,  of  a  different  sign  from  that  of  its  last  term. 

Let  the  equation  be 

xm+Axm~ "  +Bxm~2 . . .  db  U=0. 

And  first,  let  the  last  term  be  positive.  If  zero  be  put  for  x 
in  the  first  member  of  the  equation,  the  result  will  be  posi- 
tive ;  being  4-  U.  But  such  a  value  can  be  given  to  x,  that 
xm  shall  exceed  the  sum  of  all  the  other  terms.  (Art.  552.) 
Let  this  value,  which  we  will  denote  by  p,  be  taken  nega- 
tively. Then  xm  being  an  odd  power  of  x,  will  be  negative ; 
and  being  greater  than  the  sum  of  all  the  following  terms,  the 
first  member  of  the  equation  must  be  negative.  Since  then 
zero  and  —p,  when  put  for  x  in  the  first  member  of  the 
equation,  give  results  with  different  signs,  the  equation  has  a 
negative  root  between  0  and  — p.    (Art.  553.) 

Again,  let  the  last  term  be  negative.  Then  if  zero  be  put 
for  x,  the  first  member  of  the  equation  becomes  —  U,  a  nega- 
tive result.  And  by  giving  to  a;  a  positive  value  sufficiently 
large,  which  we  will  denote  by  p\  the  first  member  of  the 
equation  may  be  made  positive.  There  must  then  be  some 
positive  number  between  0  and  p'9  which  is  a  root  of  the 
equation. 

*>*>«>•  Every  equation  of  an  even  degree,  whose  last  term 
is  negative,  has  at  least  two  real  roots,  of  different  signs. 

For  when  zero  is  put  for  the  unknown  quantity  x  in  the 
first  member  of  the  equation,  the  result  will  be  negative. 
And  (Art.  552,)  if  we  give  to  a:  a  value  p  sufficiently  large, 
whether  positive  or  negative,  the  result  will  be  positive.  For 
the  first  term,  being  an  even  power  of  x,  will  be  positive, 
whether  x  is  positive  or  negative ;  and  if  such  a  value  p  is 
assumed  for  x  as  to  render  this  term  greater  than  the  sum  of 
the  others,  all  the  terms  taken  together  will  furnish  a  positive 
result.  The  equation  must  therefore  have  a  positive  root 
between  0  and  +p,  and  likewise  a  negative  root  between  0 
and  —p. 

556.  An  equation  in  which  the  signs  of  the  terms  are  all 
plus,  can  have  no  positive  root. 

For  if  a  positive  value  be  given  to  the  unknown  quantity, 
all  the  terms  in  the  first  member  of  the  equation  will  be 
positive,  and  their  sum  can  not  equal  zero. 


GENERAL     PROPERTIES     OF     EQUATIONS.       34 1 


55 7.  A  complete  equation,  in  which  the  signs  of  the  terms 
are  alternately  plus  and  minus,  can  not  have  a  negative  root 

For  then  it  would  have  a  positive  root,  if  the  signs  of  the 
alternate  terms  were  changed.  (Art.  550.)  But  this,  as  was 
shown  in  the  preceding  article,  is  impossible. 

4558.  If  an  equation  has  one  imaginary  root,  of  the  form 
a+b  y/  —  l,  it  has  another,  of  the  form  a— by/  —  1. 

For  if  a-\-by/  —  1  be  substituted  for  x  in  the  equation,  and 
the  powers  developed,  the  terms  in  which  by/  —  1  is  raised 
to  an  even  power  will  be  free  from  the  imaginary  quantity 
V  —  1  (Art.  275,)  while  those  in  which  the  odd  powers  occur 
will  contain  it.  We  may  then  write  the  result  in  this  form, 
JP+Q^  — 1  =  0;  where  P  stands  for  the  sum  of  all  the  real 
terms,  and  Q^  —  l  for  the  sum  of  all  the  terms  containing 
odd  powers  of  b.  And  this  equation  can  be  satisfied  only 
when  P=0,  and  Q=0. 

If  we  substitute  a  —  b\/  —  1  for  x,  the  result  will  not  differ 
from  the  preceding,  except  that  each  term  containing  an  odd 
power  of  b  will  have  its  sign  changed.  The  first  member  of 
the  equation  will  therefore  be  P-Qy/  —  1.  And  this  is  equal 
to  zero,  because  P=0  and  Q=0.  Hence,  a  —  bV—l  is  a 
root  of  the  equation. 

Ex.  1.  Find  the  cubic  equation,  of  which  two  of  the  roots 
are  1  and  3-f  y/ — 2.  Ans.  x3  —  7x*  +  llx—  li  =  0. 

Ex.  2.  One  root  of  the  equation 

x3—x2+2=0, 
is  —  1.     What  are  the  other  roots?  Ans.  lzhv/  —  I. 

Ex.  3.  Two  roots  of  the  equation 

x"  —  2x3  +2x2  4- 2x— 3=0, 

are  1  and  —1.     What  are  the  other  roots?  

Ans.  l=fcV—  2. 
Ex.  4,  Find  the  cubic  equation  of  which  two  of  the  roots 
are  —2  and  2—  y/ —  1.  Ans.  x3—  2a;2— 3a:+10=0. 

5«>9.  If  any  polynomial  of  the  form 

xm±:Axm- l  ±iBxm-2 dt  Txdo  U 

be  multiplied  by  a  binomial  of  the  form  x—a,  th°re  u)ill  be 
at  least  on',  more  variation  in  the  signs  of  the  product  than 
in  those  of  the  polynomial. 

29* 


342       GENERAL     PROPERTIES     OF     EttUATIONS, 

By  a  variation  is  meant  the  occurrence  of  any  sign  in  a 
polynomial,  different  from  the  preceding  sign.  When  two 
consecutive  signs  are  alike,  the  combination  is  called  a  per- 
manence. 

If,  for  example, 

X1  +2x6  +9x5  -x*  -5x*  +7x2  -4x-6, 
in  which  there  are  three  variations  of  sign,  be  multiplied  by 
x-3,  there  will  be  at  least  four  variations  in  the  product. 

The  proposition  may  be  most  easily  proved  by  attending 
first  to  a  particular  case. 

If  we  take  the  preceding  example,  and  complete  the  mul- 
tiplication as  below,  it  will  be  seen  that  there  are  six  varia- 
tions in  the  product. 

x'I+2x6+9x5—x*     -5*3+  7a?2-  4x  -  6 

x  —  3 

x*+2x'7+9x6  —  x5  —  5x*+  7x3—  4x2—  6x 

—  dx1  -6x6  -27x5  +3x*  +  \5x3  -2\x2  +  I2x+18 
xs—  xT+Hx6-28x5-2z*+22x3-25x2+  6^+18 

But  we  may  show,  without  regard  to  the  particular  values 
of  the  co-efficients,  that  there  must  be  at  least  four  variations 
in  the  signs  of  the  product,  because  there  are  three  in  the 
signs  of  the  multiplicand. 

To  do  this,  let  us  divide  the  given  polynomial  into  groups 
of  terms  having  the  same  sign,  and  pay  regard  in  multiply- 
ing, only  to  the  signs  ;   thus 


+  +  + 

—    — 

+ 

—    — 

+  +  + 

-  + 

+ 
+ 

-  4- 

+ 

_j_     #     # 

# 

+ 

# 

+ 

The  signs  of  the  terms  in  the  first  partial  product  agree 
with  the  corresponding  signs  of  the  given  polynomial ;  while 
the  terms  in  the  second  partial  product  have  the  contrary 
signs :  and  they  are  set  one  place  farther  to  the  right ;  so 
that  every  sign  in  the  first  partial  product  differs  from  the 
one  below  it,  except  at  the  beginning  of  each  group;  where 
the  upper  and  lower  signs  are  always  alike.  Now,  in  obtain- 
ing the  total  from  the  partial  products,  if  the  s:gns  of  any 
two  terms  to  be  added  are  alike,  the  sign  of  the  resulting 


GENERAL     PROPERTIES     OF     ECIUATIONS.       343 

term  must  be  the  same.  But  if  the  two  terms  have  different 
signs,  the  sign  of  the  result  may  be  plus  or  minus ;  and  can 
not  be  determined  without  having  reference  to  the  co-effi- 
cients of  the  terms.  It  appears  from  what  has  just  before 
been  stated,  that  all  the  signs  of  .the  total  product  are  thus 
ambiguous,  except  one  at  the  beginning  of  each  group.  They 
are  marked  above  by  an  asterisk. 

Now  as  the  sign  at  the  beginning  of  each  group  in  the 
multiplicand  is  the  same  as  the  sign  below  it  in  the  total  pro- 
duct, it  is  evident  that  while  the  signs  vary  from  group  to 
group  in  the  multiplicand,  there  must  likewise  be  at  least  one 
variation  from  the  first  sign  of  each  group  in  the  product  to 
the  first  sign  of  the  next  group.  And  there  is  besides  at  least 
one  variation  between  the  last  sign  in  the  product  and  the 
first  sign  of  the  preceding  group;  so  that  the  product  must 
have  at  least  one  more  variation  than  the  multiplicand. 

In  the  example  before  us,  there  is  one  variation  in  the 
multiplicand,  from  the  first  to  the  fourth  term ;  and  there  is 
at  least  one,  from  the  first  to  the  fourth  term  in  the  product. 
Again,  in  the  multiplicand,  there  is  one  variation  from  the 
fourth  to  the  sixth  term,  and  another  from  the  sixth  to  the 
seventh.  There  are  likewise  in  the  product  two  correspond- 
ing variations ;  and  there  is  one  additional  variation  from 
the  seventh  to  the  ninth  or  last  term.  The  same  reasoning, 
evidently,  may  be  applied  to  other  examples. 

In  this  demonstration,  the  polynomial  has  been  supposed  to 
be  complete.  To  render  the  demonstration  applicable  to  an 
incomplete  polynomial,  we  have  only  to  supply  the  place  of 
every  absent  term  with  ±0.  The  change  in  the  form  of  the 
polynomial  should  be  made,  without  altering  the  number  of 
the  variations.  This  may  be  done,  by  giving  to  zero,  in 
every  case,  the  sign  of  the  term  which  precedes  it.  For 
example,  the  incomplete  polynomial  x5-2xa  +  l,  should  be 
written  in  the  form  #5+0  +  0-2x,2-0  +  l;  where  the  original 
number  of  variations  is  preserved. 

We  shall  have  occasion  to  employ  the  theorem  just  proved, 
in  the  demonstration  of  the  following  proposition ;  which  is 
commonly  called  Descartes'  Rule. 

500.  The  number  of  positive  roots,  in  any  equation,  can 
not  be  greater  than  the  number  of  variations  in  the  signs  of 
its  terms.     And 

The  number  of  negative  roots,  in  any  complete  equation, 
can  not  exceed  the  number  of  permanences  in  its  signs. 


344       GENERAL     PROPERTIES     OF     EQUATIONS. 

Every  equation  having  positive  roots,  as  a,  b}  c,  &c.  may 
be  written  thus 

(zn+Hxn-*+Kxn-2  &,c.)(x-a)(x-b)(x-c)  &c.  =0; 
the  first  member  being  resolved  into  factors,  as  in  Art.  538. 
For  example, 

x5  —6x*  +  12x3  —4x2  —  13x+10=0, 
which  will  be  found,  by  trial,  to  have  the  positive  roots  1  and 
2,  may  be  written  in  the  form, 

(x'J-3x2+x+5)(x-l)(x-2)=0, 
the  polynomial  factor  being  the  product  of  all  the  binomial 
factors  of  which  the  equation  consists  (Art.  538.)  except  x-l 
and  x-2,  which  belong  to  the  positive  roots.  Now  whether 
there  be  any  variation  in  the  signs  of  the  terms  of  the  poly- 
nomial xn-\-Hxn-l-t-Kzn-2  &c.  or  not,  there  will  be  one  at 
least  after  multiplying  it  by  x- a,  two  after  multiplying  fur- 
ther by  x-b,  three  after  multiplying  by  x-c,  &c.  (Art.  559.) 
By  these  multiplications,  the  first  member  of  the  above  equa- 
tion is  reduced  to  the  form  it  must  have  had,  before  being 
resolved  into  factors.  And  we  see  that  there  must  be  at 
least  as  many  variations  in  the  signs  of  its  terms,  as  there 
are  positive  roots,  a,  b,  c,  &c. 

The  first  part  of  the  proposition  is  therefore  true.  The 
second  remains  to  be  proved. 

If  xm+Axm-l+Bxm-2  +  Cxm-*  &c.  =0 

be  a  complete  equation,  the  number  of  its  negative  roots  is 
equal  to  the  number  of  positive  roots  belonging  to  the  equation 
zm-Axm-l-{-Bxm-2  —  Cxm-3  &c.  =0     (Art.  550,) 

And  this  number  can  not  exceed  the  number  of  variations  in 
the  signs  of  the  terms  ;  as  appears  from  the  preceding  demon- 
stration. But  the  number  of  variations  in  the  second  equa- 
tion, equals  the  number  of  permanences  in  the  first.  For  it 
is  evident  that  when  two  consecutive  terms  have  unlike  signs 
in  one  of  the  equations,  the  corresponding  terms  have  like 
signs  in  the  other.  Hence  we  conclude  that  the  number  of 
negative  roots,  in  the  first  equation,  can  not  exceed  the  num- 
ber of  permanences  in  its  signs. 

It  is  evident  that  this  reasoning  will  also  apply  to  any  in- 
complete equation,  whose  terms  contain  alternately  odd  and 
even  powers  of  x ;   but  to  no  other.     Thus  it  will  apply  to 

the  equation 

n  x8-f6x5-7.z;4  -3*4-12  =  0; 


GENERAL     PROPERTIES      OF     EaUATIONS.       345 

but  not  to  the  equation 

x*—x5  +2x3  —  5x2  +7x+3=0 ; 
in  which  there  are  two  odd  powers,  Xs  and  x3,  occurring  in 
immediate  succession. 

It  is  easy  to  show,  by  an  example,  that  an  incomplete  equa- 
tion may  have  more  negative  roots  than  permanences  in  its 
signs. 

Thus  the  equation         x3  —  Ix— 6=0 
has  but  one  permanence,  while  it  has  two  negative  roots, 
—  1  and  —2. 

But  the  second  part  of  Descartes'  Rule  may  be  applied  to 
all  incomplete  as  well  as  complete  equations,  if  the  place  of 
each  deficient  term  be  first  supplied  with  ±0,  so  as  to  render 
the  equations  complete  in  form. 

Take  for  example  the  preceding  equation,  namely 
x3— Ix— 6=0; 
which,  in  its  present  form,  has  only  one  permanence.     Sup- 
plying the  place  of  the  absent  term,  we  have 
x3±0T7x-6=0, 

and  whether  +0  or  —0  be  used,  the  equation  now  has  two 
permanences ;  that  is,  it  has  as  many  permanences  as  nega- 
tive roots. 

Ex.  1.  The  equation 

x3—  2x2—  #+2=0 
has  three  real  roots.     How  many  of  them  are  positive  ? 

Ans.  Two. 
The  equation  can  not  have  more  than  one  negative  root, 
as  it  has  only  one  permanence.  And  since  the  whole  num- 
ber of  real  r^ots  is  three,  there  can  not  be  less  than  two  posi- 
tive roots.  Neither  can  there  be  more  than  two  such  roots ; 
for  the  equation  has  only  two  variations. 

Ex.  2.  The  equation 

x3+£2  +  100=0 
has  one  negative  root,  as  appears  by  Art.  554.     Is  either  of 
its  remaining  roots  negative  ? 

If  we  supply  the  place  of  the  deficient  term  with  +0,  the 
equation  becomes 

x3+a;3 +0  +  100=0. 


346       GENERAL     PROPERTIES     OF     EttUATIONS. 

Here  there  are  three  permanences ;  from  which  it  would 
appear  that  there  may  be  three  negative  roots.  But  we  are 
at  liberty  to  supply  the  place  of  the  absent  term  with  —0. 
The  equation  will  then  be 

z3+:r2-0-f-100=0; 
in  which  there  is  only  one  permanence.     And  hence  it  ap- 
pears that  there  can  not  be  more  than  one  negative  root. 

By  this  example  we  see,  that  in  supplying  the  place  of  an 
absent  term  with  ±0,  an  advantage  is  gained  by  using  the 
sign  which  will  produce  the  smallest  number  of  permanences. 

Ex.  3.  The  equation 

#4  —  5x2  +4=0 
has  four  real  roots.     How  many  of  them  are  positive  ? 
Ex.  4.    The  equation 

x5  —x*  —  5x2  —  a;-}-6=0, 
has  three  real  roots.     How  many  of  these  are  positive  ? 

56 1 .  It  has  been  remarked  in  Art.  539,  that  the  roots  of 
an  equation  are  not  always  unequal.  In  order  to  determine 
whether  a  proposed  equation  has  any  equal  roots,  we  have 
occasion  to  employ  a  property  of  derived  polynomials,  which 
we  will  here  demonstrate. 
Let  the  polynomial     . 

xm+Axm- ■  +Bxm~2 +  Tx+  U 

be  denoted  by  X;  and  let  its  first  derived  polynomial,  which  is 
mxm-l  +  (m-l)Axm-2  +  (m-2)Bxm-*...  +  T, 

be  denoted  by  X. 

This,  according  to  Art.  545,  is  the  co-efficient  of  the  first 
power  of  y  in  the  result  obtained  by  substituting  x+ y  for  x 
in  the  polynomial  X. 

Now  (Art.  538,)  X  may  be  expressed  in  the  form 
(x— a)(x—b)(x— c) .  . .  .(x—l), 

where  a,  b,  c, . . ./,  are  the  values  of  x  which  will  satisfy  the 
equation  X=0;  that  is,  are  the  m  roots  of  this  assumed 
equation.  And  if  in  this  new  form  of  X,  we  substitute  for 
x,  x+y  or  what  is  the  same  thing  y+x,  the  result  is 

(y+x  —  a)(y+x— b)(y+x— c) (y+x— I). 

The  factors  here  may  be  regarded  as  binomials,  of  which 
the  first  term  is  y,  and  the  second  terms  are  x-a,  x-bt 


GENERAL     PROPERTIES     OF     ECIUATIONS.       347 

x-c,  •••&—./.  Then  (Art.  491,)  if  the  multiplication  be  per- 
formed, the  co-efficient  of  the  first  power  of  y  in  the  result 
will  be  the  sum  of  the  products  of  the  m  quantities  x-a, 
x-b,  x-c,. .  .x-l,  taken  m-1  and  m-1.  And  it  is  evident 
that  in  order  to  obtain  these  products,  we  have  only  to  divide 
X,  in  each  case,  by  the  omitted  factor;  first  by  x-a,  next 
by  x-b,  and  so  on. 
Hence  we  infer  that 

-*T-  -**■  X  X  X  m. 

X'= +  — t  + + p   That  is, 

x— a      x  —  b      x  —  c  x—l 

A  polynomial  of  the  (m)th  degree  being  given,  its  first  de- 
rived polynomial  is  equal  to  the  sum  of  the  products  of  its 
simple  factors,  taken  m-\  and  m-1:  or  is  equal  to  the  sum 
of  the  quotients  obtained  by  dividing  the  given  polynomial 
by  each  of  its  simple  factors. 

For  example,  let  the  given  polynomial  be 
x*—  x3—7x2+x+6. 
If  this  be  made  equal  to  zero,  the  roots  of  the  equation  thus 
formed,  will  be  found,  by  trial,  to  be  1,  -1,  -2,  3.  The  sim- 
ple factors,  of  which  the  polynomial  consists,  must  therefore 
be  x-l,  x+1,  x+2,  x-3,  (Art.  538.)  Now  the  first  derived 
polynomial  of  the  given  expression  is 

4x3  —  Sx2  — 14^+1. 
And  we  shall  also  find  this  to  be  the  sum  of  the  products, 
(x+l)(x+2)(x-3),  (x-l)(x+2)(x-3),   (x-l)(x  +  l)(x-3), 

(x-l)(*+l)(a?+2), 
or  the  sum  of  the  quotients, 

JL  JL  JL    x 

x—  l'  x  +  l     r+2'   x— 3' 
where  X=x*—x3—7x2-\-x+6. 

We  may  now,  by  the  aid  of  the  preceding  theorem,  ex- 
plain the  method  by  which  the  equal  roots  of  an  equation 
are  determined. 

♦162.  Let  the  equation  be 

xm+Axm- '  +Bxm~* +  Tx+  U=0 ; 

and  let  it  have  n  roots  equal  to  a,  p  roots  equal  to  b,  q  roots 
equal  to  c,  &c.  Then  (Art.  538,)  if  the  first  member  be  de- 
noted by  X, 

X=(x-a)n(x-by(x-c)q  &c. 


348       GENERAL     PROPERTIES     OF      EQUATIONS. 

And  by  the  preceding  theorem, 

■xt.      -^         X    „  Ji.         Jl  X        X   „ 

X'= + &c.  + 7  +  r  &C.  + + &C.  +  &C. 

x-a     x—a  x-b     x-b  x-c     x-c 

the  number  of  equal  quotients  being  n  in  the  first  case,  p  in 
the  second,  q  in  the  third,  &c.  This  equation,  when  simpli- 
fied, becomes 

v        X  X         X    0 

X  =n \-p r+<7 &c. 

x—a    *  x—b     ^x—c 

Or  Xt=n(x-a)n~*(x-b)p(x-c)<!  &c. 

+p(x-a)n(x-b)p-*(x-c)*  &c. 
+q(x-a)n(x-b)p(x-cy-*   &c. 
+     &c. 
If  we  compare  the  preceding  values  of  X  and  X',  we  shall 
see  that  they  are  both  divisible  by 

(x-a)n-l(x-b)P-*(x-cy-*  &c. 

And  this  is  their  greatest  common  divisor.  For  if  not,  then 
the  quotient 

n(x-b)(x-c)kc.  +p(x-d\(x-c)  &c.  +q (x-a) (x-b)  &c.  +&c. 

which  is  obtained  by  taking  out  this  divisor  from  X,  must 
be  divisible  by  one  of  the  factors  of  X.  Now  each  of  the 
simple  factors,  x—a,  x—b,  x—c,  &c.  contained  in  X,  is  a 
divisor  of  all  the  terms  in  that  quotient,  except  one.  The 
whole  quotient  therefore  is  divisible  by  neither  of  these  factors. 

Hence, 

If  an  equation  has  equal  roots,  the  first  member  and  its 
derived  polynomial  have  a  common  divisor:  And  their  great- 
est common  divisor  is  the  product  of  all  the  factors  corres- 
ponding to  the  equal  roots,  each  raised  to  a  power  one  degree 
lower  than  in  the  given  equation. 

It  is  also  evident  that 

If  an  equation  has  no  equal  roots,  the  first  member  and  its 
derived  polynomial  have  no  common  divisor. 

For  in  that  case,  n,  p,  q,  &c.  are  each  equal  to  unity,  and 
the  expression  for  the  greatest  common  divisor  is  reduced  to 
(x—a)°(x—b)°(x—c)0  &c.,  which  is  simply  1.     (Art. 224.) 

Ex.  1.  What  are  the  equal  roots  of  the  equation 
x'  +3x*  -x3 -lx2  +  4=0  ? 


GENERAL      PROPERTIES      OF     EQUATIONS.      349 

The  derived  polynomial  of  the  first  member  is 
5x*  +  l2x3-3x2-14x; 
and  it  will  be  found  that  x2  -\-x—2  is  the  greatest  common 
measure  of  the  two  expressions.  By  solving  the  equation 
z2+x— 2  =  0,  we  obtain  the  roots,  1  and  —2.  Hence  (Art. 
538.)  x2Jrx  —  2  may  be  expressed  in  the  form  (x—l)(x+2). 
We  infer  then,  by  the  preceding  theorem,  that  the  original 
equation  contains  the  factors  x—  1,  x+2,  each  raised  to  the 
second  power;  that  is,  the  equation  has  two  roots  equal  to  1, 
and  two  others  equal  to  —2. 

If  we  divide  the  first  member  of  the  equation  by  (x  —  l)2 
(x-\-2)2,  we  shall  obtain  the  remaining  factor,  namely  x+1. 
Hence  — - 1  is  the  fifth  root  of  the  equation. 

Ex.  2.  Find  the  equal  and  the  unequal  roots  of  the  equation 
x5+4x*  —  I4x2  —  \lx  —  6=0. 

Ans.   -1,  —1,  —1;  2,  -3. 

Ex.  3.  Find  the  roots  of  the  equation 

x5+xi-\4x3+26x2-l9x+5=0. 

Ans.  1,  1,  1,  1 ;  —  5. 
Ex.  4.  Find  the  roots  of  the  equation 

xQ+2x5-12x*-Ux3+47x*  +  l2x-36=0. 

Ans.  2,  2;  -3,  —3;  1,  —1. 

563*  If  the  original  equation,  in  the  preceding  article,  be 
divided  by  the  common  measure  {x-a)n~l  (x-b)p~*  (x-c)q~l 
&c.  the  result  is  (x  —  a)(x— b)(x— c)  &c.  =0;  which  con- 
tains all  the  different  roots  of  the  original  equation,  but  no 
equal  roots. 

Sturm's   Theorem. 

o64.  This  important  theorem  receives  its  name  from  an 
eminent*  French  mathematician  by  whom  it  was  discovered 
in  1829.     It  may  be  stated  as  follows. 

Let  X  represent  the  first  member  of  an  equation  of  any 
degree ;  and  suppose  the  equation  to  have  no  equal  roots. 
Also  let  X'  be  the  first  derived  polynomial  of  X,  (Art.  545.) 
Apply  to  X  and  X'  the  rule  in  Art.  476,  for  finding  the  great- 
est common  measure ;  but  with  this  alteration,  that  the  signs 
of  each  remainder  are  to  be  changed,  before  it  is  used  as  a 
divisor.  In  other  words,  divide  X  by  X\  and  disregarding 
the  quotient,  let  the  remainder,  after  all  its  signs  are  changed, 

30 


350       GENERAL     PROPERTIES      OF     ECIUAT10N&. 

be  denoted  by  X".  Divide  X  by  X",  and  let  X"  represent 
the  remainder  with  its  signs  changed.  In  like  manner  divide 
X"  by  X"  ;  and  so  on.  As  each  successive  remainder  is 
of  a  lower  degree,  with  respect  to  x,  than  the  preceding, 
there  will  at  last  be  found  a  remainder  free  from  the  unknown 
quantity  x ;  that  is,  a  numerical  remainder.  And  this  re- 
mainder can  not  be  zero  ;  for  then  would  X  and  X  have  a 
common  divisor  (Art.  476,)  and  the  equation  X=0,  would 
have  equal  roots  (Art.  562) ;  which  is  contrary  to  the  suppo- 
sition. Let  the  last  remainder,  taken  with  the  contrary  sign. 
be  denoted  by  Xr.  Then  if  we  substitute  any  number  p  for 
x,  in  each  of  the  expressions  X,  X,  X",  X", .  ...Xr,  and 
mark  the  sign  of  the  resulting  quantity  ;  and  in  like  manner 
substitute  any  other  number  q  for  x,  and  mark  the  sign  of 
each  of  the  results ;  the  two  series  of  signs  thus  obtained, 
will  be  such  that, 

The  difference  between  the  number  of  variations  in  the  first 
series  and  that  in  the  second,  is  equal  to  the  number  of  real 
roots  of  the  given  equation,  comprized  between  p   and  q. 

o(>«».  The  demonstration  of  this  theorem  depends  on  the 
four  following  propositions. 

Prop.  I.  No  number,  substituted  for  x  in  the  series  of 
quantities  X,  X,  X" . . .  Xr,  can  reduce  any  two  consecutive 
ones  to  zero. 

If  we  take  Q',  Q",  &c.  to  represent  the  quotients  obtained 
by  dividing  X  by  X,  X  by  X"  &c.  and  remember  that  the 
remainders  of  these  divisions  are  —X",  —X",  &c.  we  shall 
have  the  following  equations, 

X  =Q'  X  -X" 

X  =Q"  X"  -X" 
X"=Q"'X"-X"f 

X<-*  =  Qr-*Xr-*-Xr 

Now  if  two  successive  quantities,  as  X'  and  X",  become 
zero,  for  any  value  of  x,  the  next  quantity  X"  must  also 
become  zero ;  as  appears  from  the  second  equation  above. 
And  since  the  values  of  X"  and  X"  are  zero,  that  of  X"" 
must  likewise  be  zero;  as  we  see  by  the  third  equation.  By 
proceeding  in  this  way,  from  one  equation  to  another,  we  at 
length  infer  that  Xr  equals  zero;  which  has  been  shown,  in 
the  preceding  article,  to  be  impossible. 


GENERAL     PROPERTIES     OF     EQUATIONS.        351 

Prop.  II.  When  one  of  the  quantities  between  X  and  Xrr 
in  the  series  X,  X,  X", . .  .Xr,  becomes  zero,  for  a  particular 
value  of  x,  the  preceding  and  the  following  quantity  must 
have  opposite  signs. 

For  if  one  of  these  intermediate  quantities,  as  X",  equals 
zero,  one  of  the  above  equations,  namely  X"=Q"X"  —  X""9 
will  become  X"=—  X"" ;  showing  that  X"  and  X""  have 
equal  values,  but  with  opposite  signs. 

Prop.  III.  If  one  of  the  quantities  between  X  and  Xr  be- 
* comas  zero  for  a  particular  value  of  x,  then  for  the  values 
of  x  a  little  greater  and  a  little  less  than  this,  the  sign  of 
that  quantity  will  form  a  permanence  with  the  sign  of  one 
of  the  two  adjacent  quantities,  and  a  variation  with  the  sign 
of  the  other. 

Suppose  that  X"=0,  when  x=a:  it  is  required  to  prove 
that  for  any  value  of  x  a  little  greater  or  a  little  less  than  a, 
the  three  consecutive  quantities  X,  X",  X",  will  have  values 
whose  signs  present  one  variation  and  one  permanence. 

Since  a  is  a  root  of  the  equation  X"=0,  it  can  not  be  a 
root  of  either  of  the  equations  X  =  0,  and  X"=0,  (Prop.  I.) 
Of  all  the  roots  of  these  two  equations,  let  that  which  is 
nearest  in  value  to  a,  be  denoted  by  b ;  and  let  h  be  any 
quantity  less  than  the  difference  between  a  and  b.  Then 
there  is  no  value  of  x  between  a+h  and  a  —  h,  which  will 
satisfy  either  of  the  two  preceding  equations.  Now  suppose 
x  to  vary  in  value  by  imperceptible  degrees  from  a— h  to 
a  +  h:  since  X  will  not  be  reduced  to  zero,  during  this  varia- 
tion, all  its  values  must  be  of  the  same  sign  ;  for  a  quantity 
varying  by  insensible  degrees,  can  not  change  its  sign  without 
becoming  zero.  All  the  values  of  X"  must  likewise  be  of 
one  sign,  while  x  varies  between  the  limits  a  —  h  and  a  +  h. 
Now  when  x=a,  X'  and  X"  have  opposite  signs,  (Prop.  II.) 
Hence,  while  x  varies  from  a— h  to  a+h,  X  has  constantly 
one  sign,  and  X"  has  the  opposite  sign.  The  sign  of  X" 
therefore,  whatever  it  may  be,  must  form  a  permanence  with 
the  sign  of  one  of  the  quantities  X  And  X",  and  a  variation 
with  the  sign  of  the  other. 

Prop.  IV.  If  (a)  is  a  root  of  the  equation  X—0,  and  any 
number  a  little  less  than  (a)  be  substituted  for  (x)  in  X  and 
X\  the  results  will  have  opposite  signs ;  but  if  a  number  a 
little  greater  than  (a)  be  substituted,  the  results  will  have  the 
same  sign. 


352  GENERAL     PROPERTIES     OF     EQUATIONS. 

Let  Z  and  Z'  represent  the  results  obtained  by  substi- 
tuting a+h  for  x  in  X  and  X.     Then  (Art.  545,) 

Z=A+A>h+~h*  +  p2hZ+  &C* 

A"' 
and  Z'=A'+A"h+—-h2  +  &c. 

where  A  and  A'  are  the  results  from  substituting  a  for  x  in 
X  and  JP,  (Art.  545) ;  and  A'  is  the  derived  polynomial  of  A. 
But  ^1=0,  since  a  is  a  root  of  the  equation  Jb=0.    Therefore* 

A"         A'" 
Z=A'h+—h2  +  —-h*  +  &c. 

Aj  Z .  o 

^//        ^/// 
=A(A'+— A+— A=+&c.) 

Now  ^1'  is  different  from  zero,  (Prop.  I ;)  and  h  may  be 
supposed  so  small  as  to  render  all  the  terms  after  A'  nearly 
equal  to  zero.  Their  sum  may  thus  be  made  less  than  A' ; 
so  that  the  sign  of  A'  shall  be  the  sign  of  the  aggregate  of 
all  the  terms.  Then  Z  will  have  the  same  sign  as  hA\  or  as 
A',  We  may  also  suppose  h  so  small  that  the  value  of  Z' 
shall  have  the  same  sign  as  the  first  term  A'.  The  signs  of 
Z  and  Z'  will  therefore  be  alike. 

But  if  a— h  be  substituted  for  x,  and  the  results  be  de- 
noted as  before,  by  Z  and  Z' ;  then 

A"         A'" 
Z=-A'h+—h2-—h*  +  &c. 

A"       A'" 
=h{-A>+  —  h-—h°+  &c.) 

and  Z>=A'-A"h+—h*-  &c. 

from  which  it  appears  that  for  small  values  of  h,  Z  has  the 
same  sign  as  h(—A'),  $r  as  —  A' ;  while  Z'  has  the  same 
sign  as  A .  In  this  case  therefore  Z  and  Z'  have  opposite 
signs. 

Thus  it  is  proved  that  if  h  be  a  very  small  quantity,  and 
a+h  be  substituted  for  a:  in  J  and  X\  the  results  will  have 
the  same  sign  ;  but  if  a—h  be  substituted,  the  results  will 
have  contrary  signs. 


GENERAL     PROPERTIES     OF     EQUATIONS.       353 

♦700.   Sturm  s  Theorem  Demonstrated. — Let  p  and  q  be 

two  numbers  between  which  are  comprized  all  the  roots  of 
the  equations 

X=0,  X'=0,  X"=0,  ...Xr-:=0; 
and  let  q  be  algebraically  less  than  p ;  that  is,  nearer  to  —  oc. 

If  q  be  substituted  for  x,  in  the  expressions  X,  X',  X", 
&c.  the  signs  of  the  resulting  values  will  present  a  certain 
number  of  permanences  and  variations.  Now  suppose  the 
value  of  x  in  these  expressions  to  begin  at  q  and  increase 
by  insensible  degrees.  The  values  of  the-  expressions  will 
thus  be  changed ;  but  they  will  have  the  same  signs  as  at 
first,  till  x  attains  a  value  which  reduces  one  of  them  to  zero. 
This  expression  will  then  change  its  sign ;  and  the  original 
series  of  signs  will  thus  be  altered.  The  vanishing  expression 
can  not  be  Xr,  since  this  is  a  number,  independent  of  x.  We 
will  first  suppose  it  to  be  X.  Then  as  the  signs  of  X  and  X 
were  unlike  before  x  reached  the  value  which  reduces  X  to 
zero,  and  are  alike  after  x  has  passed  this  value  (Prop.  IV,) 
a  variation  has  been  lost,  at  the  beginning  of  the  series  of 
signs,  by  being  changed  into  a  permanence. 

Next  suppose  either  of  the  intermediate  expressions  X, 
X",. .  .Xr~\  to  vanish  for  any  particular  value  of  x,  and  to 
change  its  sign  as  x  passes  this  value.  This  change  will  not 
affect  the  number  of  variations  in  the  series  of  signs :  for 
(Prop.  Ill,)  it  must  take  place  between  two  opposite  signs; 
and  the  three  consecutive  signs  must  form  one  permanence 
and  one  variation,  both  before  and  after  the  change.  Thus 
It  appears  that  as  x  increases  in  value,  one  variation  will  be 
lost,  by  being  changed  into  a  permanence,  whenever  x  passes 
a  root  of  the  equation  X=0]  and  in  no  other  case  will  the 
number  of  variations  be  altered  :  so  that  when  x  arrives  at 
the  value  p,  the  number  of  variations  lost,  will  be  iust  equal 
to  the  whole  number  of  real  roots  of  the  equation  X=0. 

And  from  this  demonstration  it  is  evident  that  if  p'  and  q1 
be  any  assumed  values  of  x,  the  difference  between  th^  number 
of  variations  in  the  signs  of  the  quantities  X,  X,  X" . .  ,Xr, 
for  the  value  p',  and  the  number  for  q\  is  equal  to  the  num- 
ber of  real  roots  of  the  equation  X=0,  that  are  comprized 
between  //  and  q'. 

When  the  whole  number  of  real  roots  of  an  equation  is  to 
be  determined,  we  may  take  for  p  and  q  the  values  -f  oc  and 
—  <x ;  since  every  real  root  must  be  included  between  these 
limits. 

30# 


354         GENERAL     PROPERTIES     OF     EQUATIONS 

The  number  of  positive  roots  may  be  found  by  supposing 
p=at  and  <7=0;  and  the  number  of  negative  roots,  by  sup- 
posing p=0  and  q=—oc. 

When  an  equation  has  equal  roots,  we  may  find,  by  the 
method  of  Art.  563,  another  equation  containing  all  the  dif- 
ferent roots  of  the  given  one,  but  without  the  repetition  of 
any ;  and  we  may  then  apply  the  theorem  of  Sturm  to  this 
new  equation. 

Before  proceeding  to  illustrate  the  theorem  by  examples,  it 
is  important  to  observe  that  in  deducing  X" ,  X'",  &c.  from 
X  and  X',  as  in  finding  the  greatest  common  measure  of  two 
quantities,  we  may  apply  the  principle  of  Art.  477  ;  but  the 
number  by  which  we  divide  or  multiply,  must  always  be  posi- 
tive, so  that  the  signs  of  the  quantities  X",  X",  &c.  may  not 
be  affected. 

Ex.  1.  Find  the  number  of  real  roots  of  the  equation 
x3—2x2-6x+4  =  0. 

Representing  the  first  member  by  X,  and  its  derived  poly- 
nomial by  X,  we  have 

X=Sx2  -4x-G,  (Art.  545.) 

We  now  proceed  as  in  finding  the  greatest  common  meas- 
ure of  X  and  X ;  that  is,  we  multiply  X  by  3  and  divide 
by  X,  as  follows. 


Sx3  -6x2-  lto-t-12 

Sx2—  Ax— 6 

Sx3—Ax2  —  6x 

x  -1 

(-2^2-12x  +  12) 

-3x2-18^+18 
—  Sx2-\-  4x+   6 

(-22^+12) 
-llx-f-  6 

Here  the  remainder  —  2x2  — 12#  +12  is  changed  to 
Sx2  —  \8x-\-\8,  by  dividing  it  by  +2,  and  then  multiplying 
by  +3,  (Art.  477.)  This  change  is  made,  to  enable  us  to 
carry  out  the  division.  When  the  division  is  completed,  we 
have  the  remainder  — 22.r+12.  To  simplify  this,  we  divide 
it  by  +2.  (Art.  477.)  The  result,  with  its  sign  changed  is 
llx— 6;  which  we  denote  by  X".  Proceeding  in  a  similar 
wav  with  X  and  X",  we  find  the  remainder  —441 ;  whence 
X">=+441. 


GENERAL     PROPERTIES     OF     EQUATIONS.       355 

We  have  then  this  series  of  equations 
X   =x3-2x2-Gx+4: 
X  =3x2-4x-6 
X"  =11^-6 
X'"=+441 
When  +x  or  —  oc  is  substituted  for  x,  the  signs  of  X,  X, 
&c.  must  depend  on  the  signs  of  their  first  terms,  (Art.  552.) 
Hence,  if  x=  -oc,  the  signs  of  X,  X,  X",  X",  will  be  -  +  -  +, 

forming  3  variations. 
And,  if  x=  +  oc,  the  signs       ....        will  be  +  +  +  +, 

forming  no  variation. 
The  given  equation  therefore  has  3  real  roots. 

To  determine  whether  they  are  positive  or  negative, 
Suppose  x=0:  the  signs  of  X,  X,  X",  X",  will  then  be 

H f-,  forming  2  variations. 

Hence,  one  of  the  roots  is  negative,  and  two  are  positive. 
By  assuming  different  values  for  x,  we  may  determine  very 
nearly  the  situation  of  the  roots.     Thus, 

If  x=l,  the  four  signs  will  be h+,  forming  1  variation. 

x=2,  "  --  +  +  "1 

x  =  3,  "  -  +  +  +  1 

*x=4,  "  +  +  +  +  "        0 

And  for  all  higher  values  of  x,  to  -f-x,  the  four  signs  will 
continue  to  be  positive. 

It  appears  that   one  variation   is  lost  between  x=0  and 
x=l  ;  and  another  between  x=3  and  x—4.     Hence  one  of 
the  positive  roots  lies  between  0  and  1 ;  and  the  other  be- 
tween 3  and  4.     Again, 
If  x—  —  1,  the  four  signs  will  be  +  H f-,  forming  2  variations 

x=-2,  "  ±H h       "     2  or  3     " 

x=-3,  "  — -  +  -+,       "        3 

And  for  all  higher  negative  values,  up  to  —  oc,  the  four  signs 
will  continue  to  be  the  same  as  in  the  last  case. 

Here  it  is  seen  that  when  x=—  2,  X=0  ;  from  which  it  fol- 
lows that  —2  is  the  negative  root  of  the  proposed  equation. 
We  have  given  to  X,  in  this  case,  the  ambiguous  sign  db  ; 
leaving  it  undetermined,  as  it  must  be,  whether  the  number  oi 
variations  is  2  or  3.  There  is  one  variation  gained  between 
x=  —  l  and  x——  3;   but  we  can  not  say  that  it  is  gained 


356       GENERAL     PROPERTIES     OF     EdUATIONS. 

between  x=  —  \  and  x=  —  2,  or  between  x—  —  2  and  cc=— 3: 
it  is  gained  when  x,  in  varying,  passes  through  the  value  —2* 
Ex.  2.  How  many  real  roots  has  the  equation 

2x3+2x2-8x-7=0? 
Here  we  have 

X    =2x3+2x2-8x-7 

X  =3#2+2x-4=  (6z2+4a;-8)-f-2,(Art.477.) 

X"  =  52x+55 

X"'=+7461 

When  x—-  oc,  our  series  of  signs  is  — I h ,  giving  3  variations 

"      x=  +  cc,  "  +  +  +  +,«      o 

Hence  the  equation  has  3  real  roots. 

By  assuming  different  values  for  x,  we  find  that 

When  x=  -3,  the  series  of  signs  is  — I h,  giving  3  variations 

x=-2,  "  +  +  -+       "      2 

6C=-i,  "  +-  +  +         "  2 

x=0,  "  +  +       "  1 

x=l,  "  -  +  +  +       "  1 

x=2,  *  +  +  +  +       "  0 

The  equation  therefore  has  one  negative  root  between  —2 
and  —3,  another  between  0  and  —  1,  and  a  positive  root  be- 
tween 1  and  2.  # 

Ex.  3.   How  many  real  roots  has  the  equation 

x4+4x3+x2-16x-18  =  0? 
In  this  case, 

X   =x*+4x3-\-x2-\(>x-18 

X  =2a>3;+tai+<B-8     (Art.  477.) 

X"  =5x2+25x+28 

X'"=-4to-72 

X""=-4948 

When  x=  -  oc,  the  series  of  signs  is  H h  H — ,  giving  3  variations. 

x=-2 
a?=-l 
x=0 

fc±?l 

a?=oc  " 


+-;-+- 

CI 

3 

—  ^ 

« 

2 

1 

(( 

2 

-  +  +  -- 

M 

2 

+  +  H 

M 

1 

+  +  + 

M 

1 

GENERAL     PROPERTIES     OP     EQUATIONS.       357 

The  equation  therefore  has  only  two  real  roots  ;  a  negative 
one  between  —1  and  —2,  and  a  positive  one  between  1  and 
2.     It  has  of  course  two  other  roots  that  are  imaginary. 

Ex.  4.  How  many  real  roots  has  the  equation 
£C3-f3z2+7a+4=0? 

Ans.  One  between  0  and  —1. 
Ex.  5.   How  many  real  roots  has  the  equation 

x%  —  Ax2—  8x— 4=0?  Ans.  Two. 

Ex.  6.   How  many  real  roots  has  the  equation 
2z3-5.'z-8=0? 

Ans.  One  between  2  and  3. 
Ex.  7.   How  many  real  roots  has  the  equation 

x*—7x2  —  2x4-2=0?  Ans.   Four. 

Elimination. 

567.  The  methods  of  elimination  explained  in  Sect.  VIII, 
are  particularly  adapted  to  simple  equations.  They  can  not 
often  be  conveniently  employed,  when  the  proposed  equations' 
contain  powers  and  products  of  the  unknown  quantities.  A 
method  will  now  be  given,  which  is  applicable  to  equations! 
of  every  degree. 

Let  the  proposed  equations  be 

M=0,  N=0; 
where  M  and  N  stand  for  any  polynomials  containing  x  and 
y.  If  we  apply  to  M  and  N  the  process  for  finding  their 
greatest  common  measure,  (Art.  476,)  and  denote  the  succes- 
sive quotients  by  Q,  Q\  &c.  and  the  remainders  by  R,  R', 
&c.  then  since 

M=NQ+R,  N=RQ+R',  R=R'Q"+R",  dec. 
and  since  M=0,  and  iV=0,  we  see  that  R  and  each  of  the 
following  remainders  must  be  equal  to  zero.  Now  if  the 
polynomials  M  and  N  have  been  arranged  according  to  the 
powers  of  xf  each  remainder  must  be  of  a  lower  degree, 
with  respect  to  this  quantity,  than  the  preceding  one;  and 
there  must  finally  be  a  remainder  which  is  entirely  free  from 
x.  If  we  make  this  remainder  equal  to  zero,  we  shall  have 
an  equation  which  contains  only  the  unknown  quantity  y, 
and  which  has  been  obtained  by  eliminating  x  from  the  two 
given  equations. 


358  GENERAL     PROPERTIES     OF     EQUATIONS. 

The  value  of  x  may  be  obtained,  in  terms  of  y,  by  making 
the  last  remainder  but  one  equal  to  zero. 
Ex.  1.  Eliminate  x  from  the  equations 
4x3  —  y2x— y2  — 1  =  0, 
2x2+yx— a=0. 
Applying  the  method  just  explained,  we  proceed  as  follows. 


4x3  —  y2x— y2  —  1 
4x3+2yx2—  2ax 


2x2+yx—a 


2x-y 


—2yx2  +  (2a-y2)x—y2  —  l 
—2yx2—y2x  -\-ay 

2ax—y2—ay—\  —  1st  remainder. 
2ax—y2  —ay— I 


2ax2  +ayx—a2 

2ax2-(y2+ay+l)x      x+y2+2ay+l 

(y2+2ay+\)x-a2 
2a(y2  +2ay+l)x-2a3 
2a(y2  +2ay  +  l)x-(y2  +ay+l)(y2+2ay+l) 

~(y2~+^y+lj(y2+2ay+l)-2a3  =  2d  rem. 
Ans.  y*+3ay*+2(a2+  l)y2 +3ay-2a3  +  l=0. 

The  value  of  x,  in  terms  of  y,  is — 

2a 

For  the  sake  of  avoiding  fractions,  the  partial  remainder 
(y2+2ay+l)x  —  a2   is  here  multiplied  by  2a,  (Art.  477,)  be- 
fore completing  the  division.     The  first  divisor  also,  before 
being  divided  by  the  first  remainder,  is  multiplied  by  a. 
Ex.  2.  Eliminate  x  from  the  equations 
x2-\-y— 1  =  0 
2xy— y— 1=0 

Ans.  4?/3-3?/2+2y+l  =  0. 
Ex.  3.  Eliminate  x  from  the  equations 
2x2  —  xy2+y3  — 1  =  0 
x2—xy  +*/a  — 1  =  0 
Ans.  y6—  4z/5+3y4+5?/3  —  6y2  +  l=0. 
Ex.  4.  Eliminate  x  from  the  equations 
x3+y3  —  a=0 
.T2+^//+?/a-6=0 
Ans.  4y6  —  GbyA—4ay3+3b2y2+3aby+a2—b:i==0. 


GENERAL     PROPERTIES     OF     ECIUATIONS.       359 

Ex.  5.  Eliminate  x  from  the  equations 
x2+ay—b=0 
y2+cx—d=0 

Ans.  y4  —  2dy2+ac2y+d2  —  bc2=0. 
When  we  have  three  equations  containing  three  unknown 
quantities,  we  may  first  eliminate  one  of  the  quantities,  by 
combining  either  of  the  equations  with  the  other  two ;  and 
having  thus  obtained  two  new  equations  containing  only  two 
unknown  quantities,  we  may  proceed  with  these  as  in  the 
former  case. 

A  similar  method  may  be  pursued  in  cases  where  there  are 
four  or  more  equations  given. 

Ex.  6.  Eliminate  x  and  y  from  the  equations 
xy  —  z— 10=0 
xz—y  +  l\=0 
yz—x-{-l4=0 
Ans.  z5-fl0z4-2%3-174%2-316z- 144=0. 
Ex.  7.  Eliminate  x  and  y  from  the  equations 
x-\-xy— a=0 
y-\-yz  —  b  =  0 
z  +  zx— c=0 

Here  we  may  eliminate  x  by  combining  the  first  and  third 
equations  ;  and  we  must  then  combine  the  new  equation  with 
the  second,  in  order  to  eliminate  y. 

Ans.  (a+l)z2  +  (a+b-c+l)z-c(b+l)=0. 

Ex.  8.  Eliminate  x  and  y  from  the  equations 

y(x+y-z)  =  a 

z(y-\-z—x)=b 

x(z+x—y)=c 
Ans.  8z8-4(a+56+2c)z6+2(2^+9ft2-}-5^+c2)z4 

-b2(a+7b  +  3c)z2+b4=0. 


3G0  RESOLUTION      OF      EQUATIONS 


SECTION    XX. 


RESOLUTION"    OF    EQUATIONS. 

Art.  068.  The  real  roots  of  equations  are  either  rational 
or  irrational.  Rational  or  commensurable  roots  are  such  as 
can  he  exactly  expressed  in  numbers.  Irrational  or  incom- 
mensurable roots  are  those  which  can  only  be  expressed  ap- 
proximately. 

We  will  first  explain  the  method  of  finding 

Rational  Roots. 

569.  An  equation  whose  co-efficients  are  whole  numbers, 
and  that  of  the  first  term  unity,  can  have  only  whole  numbers 
for  its  rational  roots. 

For  let  the  equation  be 

xm+Axm~  ■  +  Bxm~2  . . .  +  Tx+  U=0 ; 
where  A,  B,. .  mT,  U,  are  whole  numbers ;  and  if  possible  let 

the  irreducible  fraction  7  be  a  root  of  the  equation. 

am        am~l         am~2  a 

Then  (Art.  537,)  -+^_+£_. .  .+T^  +  U=0. 

If  we  multiply  this  by  bm~l  and  transpose,  we  obtain 

am 

-r  =  ~  Aam~ J  -  Bam~2b ...  -  Tabm~2  —  Vbm' l . 
0 

am 
Here  the  first  member  -7-  is  an  irreducible  fraction :  for 
0 

a 

since  7   is  in  its  lowest  terms,   a  and  b  have  no  common 

divisor ;  and  therefore  am  and  b  can  have  no  common  divi- 
sor. But  the  second  member  is  a  whole  number,  since  all  its 
terms  are  integral.  We  have  then  an  irreducible  fraction 
equal  to  a  whole  number ;  which  is  absurd. 

•570.  We  proceed  therefore  to  show  how  the  integral 
roots  of  an  equation  may  be  determined,  supposing  the  co- 


RESOLUTION      OF      EQUATIONS  361 

efficients  as  before  to  be  whole  numbers,  and  that  of  the  first 
term  unity. 

Let  the  given  equation  be 

x*+Ac3  +  Br2  +  Cx-\-D=0. 
From  Art.  538,  it  appears  that  if  any  number  a  be  a  root  of 
this  equation,  the  first  member  can  be  resolved  into  two  fac- 
tors, one  of  which  is  x  —  a,  and  the  other  is  a  polynomial  of 
the  third  degree.     Let  this  bd  represented  by 

x*  +  Ax2+B'x  +  C. 
Then  x*+Ax*  +  Bx2  +  Cx+ D=  {x-a)(x2  +  A'x2+B'x+Q) 
=x*+(A'-a)x>  +  (B,-aA')x2+(Cf-aB')x-aC' 
As  the  two  members  of  this  equation  must  be  precisely 
alike,  we  see  that       (1)     A'— a     —A 

(2)  B'-aA=B 

(3)  C'-aB'=C 

(4)  -aC'  =  D 

To  obtain  now  the  integral  roots  of  the  given  equation; 
we  have  only  to  find  every  integral  value  of  a,  which  will 
satisfy  these  four  conditions. 

Since  A,  B,  C  and  D  are  supposed  to  be  integers,  it  is  evi- 
dent from  the  equations  (1),  (2),  (3),  that  if  a  be  an  integer, 
A',  B',  C,  are  also  integers,  Now  the  four  equations  of 
condition,  taken  in  the  contrary  order,  may  be  changed  into 
the  following. 

a 

a 
B-B' 


a 

A-A' 

-1 


a 

Whence  it  appears  that 

The  quotient  of  D  divided  by  a  must  be  a  whole  number. 

And  if  this  quotient  be  added  to  C,  and  the  sum  divided 
by  a,  the  new  quotient  must  be  a  whole  number. 

And  again,  if  this  be  added  to   B,  and  the  sum  divided  by 
a,  the  quotient  must  be  a  whole  number. 

And  finally,  if  this  be  added  to  A,  and  the  sum  divided  by 
a,  the  quotient  must  be  — -1. 

31 


862  RESOLUTION      OF      EQUATIONS. 

It  follows  from  the  first  of  these  conditions,  that  all  the 
integral  roots  of  the  given  equation  are  comprized  among 
the  divisors  of  its  last  term. 

If  then  we  substitute  each  of  these  divisors  for  a,  and 
ascertain  by  trial  which  of  them  will  satisfy  the  above  con- 
ditions, the  required  roots  will  be  determined. 

But  after  obtaining,  in  this  manner,  one  integral  root,  as  a, 
we  may  depress  the  given  equation  to  a  lower  degree,  by 
dividing  by  x-a.  The  result  will  be  x2-\-A'x2-\-B'x-\-C'=0. 
Now  the  values  of  —  C,  —  B',  —  A',  were  found  in  testing 
the  root  a;  so  that  no  actual  division  is  required,  to  obtain 
this  new  equation.  The  roots  of  the  cubic  equation  are  the 
same  as  those  of  the  original  equation,  with  the  exception  of 
a.  (Arts.  535,  538.)  Then  instead  of  seeking  a  second  inte- 
gral root  of  the  given  equation,  we  may  proceed  to  find  an 
integral  root  of  the  new  equation ;  which  is  a  simpler  prob- 
lem. The  same  method  may  be  pursued  in  finding  all  the 
other  integral  roots. 

The  method  which  has  now  been  explained  with  reference 
to  an  equation  of  the  fourth  degree,  is  obviously  applicable 
to  equations  of  other  degrees.  Hence  we  have  the  following 
general 

Rule. 

To  find  the  integral  roots  of  any  equation,  in  which  the 
co-efficients  are  integers,  and  that  of  the  first  term  unity, 

Divide  the  last  term  of  the  equation  by  one  of  its  factors, 
which  may  be  denoted  by  (a) ;  add  the  result  to  the  co-efficient 
of  x,  the  unknown  quantity,  and  divide  the  sum  by  (a).  The 
quotient  should  be  a  whole  number.  If  so,  add  it  to  the  co- 
efficient of  x2,  and  divide  the  sum  by  (a).  If  the  quotient  is  a 
whole  number,  add  it  to  the  co-efficient  of  x3  and  divide  by  (a). 
And  so  proceed  till  all  the  co-efficients  of  the  equation  have 
been  employed,  except  that  of  thefrst  term.  The  last  quotient 
should  be  -1.  If  this  is  the  case,  the  number  (a)  must  be  a 
root  of  the  equation.  But  if  (a)  fails  to  render  any  of  the 
quotients  integral,  or  to  moke  the  last  quotient  — -1,  it  can 
not  be  a  root,  and  a  new  number  must  be  tried. 

Having  obtained-  one  intgral  r<*ot,  to  find  another,  change 
the  signs  of  the  quotients  obtained  in  trying  the  root  already 
found ;  then  make  these  quotients  the  co-efficirnts  of  an  equa- 
tion of  the  next  lower  degree ;  and  proceed  with  this  as  with 
the  original  equation.  And  in  the  same  manner  find  all  the 
remaining  integral  roots. 


RESOLUTION      OF      EQUATIONS.  363 

If  any  term  is  wanting  in  the  given  equation,  zero  must  be 
used  in  the  place  of  its  co-efficient. 

It  is  generally  best  to  try  the  smaller  divisors  first,  begin- 
ning  always  with  1  and  — 1. 

When  a=l,  the  above  equations  of  condition  may  be  re- 
duced to  these. 

A'=A  +  1 
B'=B+A' 
C'=C+B' 
D+C'=0,  or  D=-C. 
And  when  a—  —  1, 

A=A-l 
B'=B-A' 
C=C-B' 
D-C=0,  or  D=C 

.  These  formulas,  with  the  corresponding  ones  for  equations 
of  other  degrees  than  the  fourth,  are  more  convenient  than 
the  preceding  rule,  for  trying  the  particular  divisors  1  and 
-1. 

Ex.  1.  Find  the  integral  roots  of  the  equation 

x5  -6x*  —2x3  +36x2  +.r-30  =  0. 
Trying  the  divisor  1  by  the  method  just  indicated,  we  find 
j4'=  —  6+  1  =  — 5 
B'  =  -  2-  5=-7 
C'=  36-  7=  29 
D'=  1-1-29=  30 
-30  +  Z>'=-30-f30  =     0. 

Here  we  have  jive  equations  of  condition,  because  the 
given  equation  is  of  the  fifth  degree ;  and  as  they  are  all  sat- 
isfied, 1  is  a  root  of  this  equation. 

If  the  given  equation  be  divided  by  x—l,  the  result  will  be 

x*  -5x3-7x2  +29^  +  30=0 ; 

in  which  the  co-efficients  are  the  values  of  A',  B'f  C,  D'}  as 
found  above. 

Trying  the  divisor  1  in  this  equation,  we  find 
4'=-4,  £'=-11,   C'=18,    D-fC'=30+18=48.     And  as 
the  sum  of  D  and  C  is  not  zero,  1  can  not  be  a  root. 


RESOLUTION      OF      EQUATIONS. 

Trying  —  1  we  find 

J'=-5-l  =  -6 
jB'  =  -7+6=-1 
C'=  29+1=  30 
D-C'=  30-30=  0 
Therefore  —  1  is  a  root ;  and  the  next  lower  equation  is 
x3  —  6x2—  x+30=0. 
Here  if  the  divisor  —  1  be  tried,  it  will  be  found  unsatisfactory. 
Trying  2,  which  is  a  divisor  of  30,  by  the  preceding  rule, 
we  obtain 

30 

-1  +  15 

— =7 

-6+71 
2     ~V 

As  the  last  quotient  is  not  —  1,  2  is  not  a  root. 

If  we  try  —2,  we  shall  find  the  successive  quotients  to  be 
—  15,  8,  —  I.  Therefore  —2  is  a  root;  and  the  next  lower 
equation  is  x2  —  te+15=0. 

The  two  remaining  roots  are  3  and  5 ;  as  may  be  found  by 
resolving  this  quadratic  in  the  usual  way. 

Ex.  2.  Find  the  roots  of  the  equation 

#4— 81x'2— 310x—  150=0. 

Here  the  divisors  of  the  last  term  are  1,  - 1,  2,  -2,  3,  -3,  5,  -5, 

&c.   ■  It  will  be  seen  by  trial  that  none  of  the  first  seven  of 

these  is  a  root.     But  if  we  try  —5,  we  obtain 

-150 
-=30 

-310  +  30 
-^--  =  56 

-81+56 


-5 

0   +5 


=  -1. 


.      -5 

Therefore  —5  is  a  root  of  the  given  equation;  and  the 
next  lower  equation  is 

<c3  — 5x2  —  56.x— 30=0. 


RESOLUTION      OF      ECIUATIONS.  365 

U  we  again  try  —  5,  which  is  a  div;sor  of  —30,  we  shall 
find  it  to  be  a  root  of  this  equation  ;  and  the  next  lower  equa- 
tion will  be 

x2  —  \0x— 6=0; 

of  which  the  roots  are  5:fc>/31.     The  roots  of  the  given 
equation  therefore  are  —5,  —5,  5z±rv/31. 

Ex.  3.  Find  the  roots  of  the  equation 

x'  -30x3  +869x-30=0. 

Ans.   -5,  6,  }(29=h>/837). 
Ex.  4.  Find  the  roots  of  the  equation 

x5  -2xA  —  40x3  -  10x2  +279u:+252=0. 
Ex.  5.  Find  the  roots  of  the  equation 

x* -8x5  +6x*  +C>\x3 -VOx2  -  116x+80=0. 
Ex.  6.  Find  the  roots  of  the  equation 

x1-x6  —  14x5  4-  14x4-f-49.r3  — 40r3  -36x4-36=0. 
Ex.  7.  Find  the  roots  of  the  equation 

x*+4x*  —  36x2-37x— 40=0. 
Ex.  8.  Find  the  roots  of  the  equation 

x5  -32x3+6x2  +  \lbx-  150=0. 

Solution  of   Equations   bv  Methods  of  Approximation. 

«571.  It  will  genera-lv  be  best,  to  determine  the  rational 
roots  of  numerical  equations  by  the  preceding  method.  Bui 
rational  and  irrational  roots  may  a!I  be  found  with  sufficient 
exactness  by  success" ve  approximations.  The  following 
methods  are  most  commonly  employed  for  this  purpose. 

Approximation  by  Double  Position. 

572.  From  the  laws  of  the  co-efficients,  as  stated  in 
Art.  511,  a  general  estimate  may  be  formed  of  the  values  of 
the  roots  of  anv  equation.  They  must  be  such,  that,  when 
their  signs  are  channel,  their  product  shall  be  equal  to  the  last 
term  of  the  equation!  an  1  ther  sun  equal  to  the  co-effieient 
of  the  s°conl  t*rm.  A  trial  may  th  n  be  male,  bv  substi- 
tuting in  thi  place  or  t'l  j  unknown  'etter,  its  supposed  value. 
If  th  s  proves  to  be  too  small  or  too  great,  it  miy  be  increased 
or  diminished,  an  1  the  trials  repeated,  till  one  is  found  which 
will  nearly  satisfy  the  conditions  of  the  equations.     After  we 

31* 


BGQ  RESOLUTION      OF      EQUATIONS. 

have  discovered  or  assumed  two  approximate  values,  and  cal- 
culated the  errors  which  result  from  them,  we  may  obtain  a 
more  exact  correction  of  the  root,  by  the  following  proportion. 
As  the  difference  of  the  errors,  to  the  difference  of  the  as- 
sumed numbers ; 

So  is  the  least  error,  to  the  correction  required,  in  the  cor- 
responding assumed  number. 

This  is  founded  on  the  supposition,  that  the  errors  in  the 
results  are  proportioned  to  the  errors  in  the  assumed  numbers. 
Let    N  and  n  be  the  assumed  numbers ; 

$    and   s,       the  errors  of  these  numbers  ; 
R   and  r,       the  errors  in  the  results. 
Then  by  the  supposition,  R  :  r  : :  S :  s 

And  subt.  the  consequents  (Art.  397,)    R—r  :  8—5  :  :  r  :  s. 

But  the  difference  of  the  assumed  numbers  is  the  same  as 
the  difference  of  their  errors.  If  for  instance,  the  true  num- 
ber is  10,  and  the  assumed  numbers  12  and  15,  the  errors  are 
2  and  5;  and  the  difference  between  2  and  5  is  the  same  as 
between  12  and  15  Substituting,  then,  N—n  for  S—s,  we 
have  R-r  :  N-n  :  :  r :  5,  which  is  the  proportion  stated  above. 

The  term  difference  is  to  be  understood  here,  as  it  is  com- 
monly used  in  algebra,  to  express  the  result  of  subtraction 
according  to  the  general  rule.  (Art.  75.)  In  this  sense,  the 
difference  of  two  numbers,  one  of  which  is  positive  and  the 
other  negative,  is  the  same  as  their  sum  would  be,  if  their 
signs  were  aiike      (Art.  78.) 

The  supposition  which  is  made  the  foundation  of  the  rule 
for  finding  the  true  value  of  the  root  of  an  equation,  is  not 
strictly  correct.  The  errors  in  the  results  are  not  exactly 
proportioned  to  the  errors  in  the  assumed  numbers.  But  as 
a  greater  error  in  the  assumed  number,  will  generally  lead 
to  a  greater  error  in  the  result,  than  a  less  one,  the  rule  will 
answer  the  purpose  of  approximation.  If  the  value  which  is 
first  found,  is  not  sufficiently  correct,  this  may  be  taken  as  one 
of  the  numbers  for  a  second  trial;  and  the" process  may  be 
repeated  till  the  error  is  diminished  as  much  as  is  required. 
There  will  generally  be  an  advantage  in  assuming  two  num- 
bers whose  difference  is  -1,  or  -01,  or  *001,  &c. 

Ex   1.  Find  the  value  of  x,  in  the  cubic  equation 

x*  —  8x2  +  nx— 10=0. 


RESOLUTION      OF      ECIUATIONS.  367 

Here  as  the  s'gns  of  the  terms  are  alternately  positive  and 
negative,  the  roots  must  be  all  positive;  (Art.  542,)  their  pro- 
duct must  be  10  and  their  sum  8. 

Let  it  be  supposed  that  one  of  them  is  51  or  52.  Then, sub- 
stituting these  numbers  for  x,  in  the  given  equation,  we  have 
By  the  1st  suppos'n,  (51)3-8X(51)2  +  17X(51)-10=1*271 
By  the  second,  (5*2) 3 -8  X  (5'2) 2 +  17  X  (5  2) -10=2688 

That  is,    By  the  first  supposition,    By  the  second  supposition, 


The  1st  term,             x3  =     132  651 

140608 

The  2d,                  -8.r2  = -208*08 

-21632 

The  3d,                     17.r   =       867 

88-4 

The  4th,               -10     =  -    10- 

-    10* 

Sums  or  errors,                   +1*271 

+2*688 

Subtracting  one  from  the  other, 

1271 

Their  difference  is  1417 

Then  stating  the  proportion 

1-4:01  ::  1-27:0*09, 
the  correction  to  be  subtracted  from  the  first  assumed  num- 
ber 51  :    The  remainder  is  501,  which  is  a  near  value  of  x. 

To  correct  this  farther,  assume  ^=5*01,  or  502. 

By  the  first  supposition.         By  the  second  supposition. 
The  1st  term,  x3=     125751  126*506 

The  2d,  -8^2  =  -2008  -201*6 

The  3d,  m=       8517  85  34 

The  4th,  —io=-    10-  -    10* 

Errors,  +     0121  +     0246 

0J21 

Difference,  01 25 

Then  0125  :  001  :  :  0121  :  0  01,  the  correction.     This  sub- 
tracted from  5*01,  leaves  5  for  the  value  of  x ;  which  will  be 
found,  on  trial,  to  satisfy  the  conditions  of  the  equation. 
For  53-8X52  +  17X5-10=0. 

We  have  thus  obtained  one  of  the  three  roots.  To  find 
the  other  two.  let  the  equation  be  divided  bv  x  —  5,  accord- 
ing to  Art.  126,  ani  it  will  be  depressed  to  the  next  inferior 
degree.     (Art.  538.) 

x-5)x3  -Sx2  +  17x- 10(^2  -3.r+2=0. 


368  RESOLUTION      OF      EQUATIONS. 

Here  the  equation  becomes  quadratic. 

By  transposition,  #3— 3.r=— 2 

Completing  the  square,  x2  —  3x+|  =  |— 2=J 

Extracting  and  transposing,  x=l  =fc  y/i  =  % zb£. 

The  first  of  these  values  of  x,  is  2,  and  the  other  1. 

We  have  now  found  the  three  roots  of  the  proposed  equa- 
tion. When  their  signs  are  changed,  their  sum  is  —8,  the 
co-efficient  of  the  second  term,  and  their  product  —10,  the 
last  term. 

2.  What  are  the  roots  of  the  equation 

xz—$x2  +4-z-J-48=0?  Ans.   —2,  +4,  +6. 

3.  What  are  the  roots  of  the  equation 

;r3-16.z;2+65.r-50=0?  Ans.   1,  5,  10. 

4.  What  are  the  roots  of  the  equation 

^3  +  2^2 -33^=90?  Ans.  6,-5,  -3. 

5.  What  is  a  near  value  of  one  of  the  roots  of  the  equation 

:r3+9.r2+4.r=80? 

6.  What  is  a  near  value  of  one  of  the  roots  of  the  equation 

x2+x2+x=\W? 

Newton's  Method. 

•573-  A  second  method  of  approximating  to  the  roots  of 
numerical  equations,  is  that  of  Newton,  by  successive  sub- 
stitutions. 

Let  r  be  put  for  a  number  found  by  trial  to  be  nearly  equal 
to  the  root  required,  and  let  z  denote  the  difference  between  r 
and  the  true  root  x.  Then  in  the  given  equation,  substitute 
r±z  for  x,  and  reject  the  terms  which  contain  the  powers  of  z. 

This  will  reduce  the  equation  to  a  simple  one.  And  if  z 
be  less  than  a  unit,  its  powers  will  be  still  less,  and  therefore 
the  error  occasioned  by  the  rejection  of  the  terms  in  which 
they  are  contained,  will  be  comparativelv  small.  If  the 
value  of  z,  as  found  by  the  reduction  of  the  new  equation. 
be  added  to  or  subtracted  from  r,  according  as  the  latter  is 
found  by  trial  to  be  too  great  or  too  small,  the  assumed  root 
will  be  once  corrected. 

By  repeating  the  process,  and  subst;tutinor  the  corrected 
value  of  r\  for  its  assumed  value,  we  may  come  nearer  and 
Hearer  to  the  root  required. 


RESOLUTION      OF      EdUATIONS.  369 

Ex.  1.  Find  one  of  the  values  of  x,  in  the  equation 
x3-16.r2+65^=50. 
Let  r—  z=x. 

/  z3  =  (r—z)2=r3—3r2z+3rz2—z3  \ 

Then    <  -I6x2  =  -16(r-z)2  =  -l6r2  +32rz-16z2    >  =50. 

(      65r  =     65(r— z)   =     65r  —  65z  ) 

Rejecting  the  terms  which  contain  z2   and  z3,  we  have 

r3-16r2+65r-3r2z-r-32rz  — 65z  =  50. 

50  —  r3-f-16r2  —  65r 


This  reduced  gives  z= 


—  3r2+32r— 65 


If  r  be  assumed  =11,  then  z=-— =0*8  nearly. 

and     x—r—  z  nearly  =11  —  0  8=  10*2. 
To  obtain  a  nearer  approximation  to  the  root,  let  the  cor- 
rected value  of  10  2  be  now  substituted  for  r,  in  the  preceding 
equation,  instead  of  the  assumed  value  11,  and  we  shall  have 
z=188  x=r—z  =  10012. 

For  a  third  approximation,  let  r=  10*012,  and  we  have 
z=012  x—r— z=10. 

2.  What  is  a  near  value  of  one  of  the  roots  of  the  equation 

.z3-fl0x2+5.£=2600?  Ans.   110067. 

3.  What  are  the  roots  of  the  equation 

x*+2x2  —  llx— 12? 

4.  What  are  the  roots  of  the  equation 

x*  +4x3  —  lx2  —  34x=24  ? 

Horner  s  Method. 

•57  4.  Another  convenient  method  of  approximating  to  the 
roots  of  equations  was  first  published  by  Mr.  Horner  in  1819. 

To  explain  this,  we  will  suppose  the  given  equation  to  be 
2z'4-5.r3+3^2—  x  — 4  =  0. 
By  Sturm's  Theorem,  or  by  trial,  according  to  Art.  553,  we 
easily  find  that  2  is  the  first  figure  of  one  of  its  roots.  To 
find  the  remaining  figures,  let  the  equation  be  transformed 
into  another,  whose  roots  are  less  than  those  of  the  given  one 
by  2.     The  result  will  be 

2y4  +  lly3+2l3/3  +  15?/-2=0.     (See  Art.  518,  Ex.1.) 


370  RESOLUTION     OF     ECIUATIONS. 

As  y  is  less  than  x  by  2,  it  is  the  decimal  part  of  x\  and 
the  first  figure  of  y  is  the  second  figure  of  x.  By  Art.  553, 
we  find  this  figure  to  be  .1.  Let  us  now  transform  the  last 
equation  into  another  whose  roots  are  less  than  its  own  by  .  1. 
The  new  equation  will  be 

2%4-fll.8z3-f24.42z2+19. 538z-.  2788=0.   (Art.  548,  Ex.2.) 

The  first  figure  of  its  root  we  shall  find  to  b3  .01.  This 
therefore  is  the  second  figure  in  the  value  of  y,  and  the  third 
in  that  of  x.  By  proceeding  thus,  we  can  determine  the  root 
of  the  given  equation  to  any  required  number  of  figures. 

This  method  would  be  very  inconvenient,  if  we  were 
obliged  to  find  all  the  figures  of  the  required  root  by  trial. 
But  after  obtaining  one  or  two  of  them  in  this  way,  the  oth- 
ers may  be  found  in  a  much  simpler  manner.  Thus  the 
equation  containing  z  may  be  written  in  the  form 

.2788 

Z~  19. 538+24. 42z  +  11.8z2+2z3 
Now  the  value  of  y  was  found  to  be  between  .1  and  .2. 
Therefore  z,  which  equals  y—  .1,  must  be  less  than  .1  ;  and 
the  higher  powers  of  z  are  still  smaller.  All  the  terms  after 
19.538,  in  the  above  denominator,  are  therefore  small,  and 
the  value  of  the  whole  fraction  can  not  differ  much  from 

»  that  is  .0142.     Hence,  z  must  be  very  nearly  equal 

to  the  latter  quantity.  And  as  we  require  only  the  first  figure 
in  the  value  of  z,  we  may  assume  this  to  be  .01,  without 
danger  of  error. 

If  other  equations  were  obtained  by  transformation,  their 
roots  would  ha  continually  decreasing,  and  the  first  figure  of 
each  miirht  be  found,  as  in  the  case  just  considered,  by  divid- 
ing the  final  term  of  the  equation,  taken  with  a  contrary  sign, 
by  the  co-efficient  of  the  last  term  but  one. 

From  what  has  now  been  shown,  we  derive  the  following 
rule  for  obtaining  a  root  of  an  equation  of  any  degree. 

Find  by  Sturm's  Theorem  or  by  trial,  the  first  figure  of 
the  root. 

Transform  the  equation  into  another  whose  roots  are  less 
than  those  of  the  former,  by  the  figure  just  found. 

Divide  the  last  term  of  the  new  equation,  taken  with  a  con- 
trary  sign,  by  the  co-efficient  of  the  term  before  the  last ;  to 
obtain  a  second  figure  of  the  required  root. 


RESOLUTION      OF      EaUATIONS.  37 1 

Then  transform  the  second  equation  into  another  whose 
roots  are  less  than  its  own,  by  the  figure  last  found : 

Divide  as  before,  to  obtain  a  third  figure :  and  so  proceed 
till  the  required  number  of  figures  have  been  obtained. 

It  will  sometimes  be  necessary  to  find  the  second  figure  of 
the  root,  as  well  as  the  first,  by  trial.  In  Ex.  3  below,  even 
the  third  figure  must  be  found  in  this  way. 

The  manner  of  applying  the  rule  will  be  shown  by  a  few  ex- 
amples. And  we  will  begin  with  the  equation  employed  above. 

Ex.  1.  Find  a  root  of  the  equation 

2x*  -5x3  +3x2  —x—4  =  0. 

The  first  figure  of  the  required  root,  as  found  by  trial,  is  2. 
To  find  other  figures,  we  proceed  as  follows. 


2 

-  5 

3 

-  1        -4  (2.114 

-  1 

1 

1       i  -2 

3 

7 

il5       2-  .2788 

7 

i21 

17.212   a-  .08096618 

ill. 2 

22.12 

2  19.538 

11.4 

23.26 

19.783382 

11.6 

224.42 

320.029948 

all. 82 

21.5382 

11.84 

24.6566 

11.86 

324.7752 

3II.88 

Here  the  co-efficients  of  the  equation,  obtained  by  trans- 
forming the  given  one  as  in  Art.  548,  Ex.  1,  are  indicated  by 
the  small  figure  i,  placed  at  the  left  of  each;  with  the  excep- 
tion of  the  first  co-efficient  2,  which  is  the  same  as  that  of 
the  primitive  equation.  Dividing  2,  that  is,  —2  taken  with 
a  contrary  sign,  by  15,  we  obtain  .1  for  the  second  figure  of 
the  required  root.  Then  from  the  co-efficients  marked  r,  we 
obtain  those  of  the  next  equation,  which  are  marked  2.  (See 
Art.  548,  Ex.  2.)  Again,  dividing  .2788  by  19.538,  we  find 
.01  for  the  third  figure  of  the  root.  And  finally,  by  obtain- 
ing the  co-efficients  of  the  next  equation,  and  dividing  as 
before,  we  find  .004  for  the  fourth  figure  of  the  root. 

It  must  be  observed  that  the  mark  1  in  the  first  column  of 
numbers,  applies  not  to  the  whole  of  the  number  11.2,  but  to 
11  only.  For  the  sake  of  brevity,  the  decimal  .2  is  added  at 
the  right  of  11,  to  form  the  following  number  11.2;  instead 
of  writing  the  two  numbers  separatelv.  In  the  same  column, 
the  mirk  2  applies  only  to  the  first  three  figures  of  the  num- 
ber 11.82. 


372  RESOLUTION      OF      EQUATIONS. 

Ex.  2.  Find  a  root  of  the  equation 
x3-4x2  —  4^+20=0. 

One  of  its  roots  is  between  3  and  4.  (Art.  553.)    For  obtain- 
ing this,  the  process  is  as  follows. 

1      -4  -4  20   (3,525 

-1  -7  i-    1 

2  i~l  2-      .125 

i5.5  1.75  s-      .027392 

6.0  2     4.75  4—      .002171875 

26.52  4.8804 

6.54        s     5.0112 
36.565  5.044025 

6.570      4     5.076875 
46.575 

The  first  four  figures  of  the  root  are  therefore  3.525. 

57o.  When  several  figures  of  a  root  have  been  compu- 
ted as  in  the  preceding  examples,  a  number  of  others  may 
be  obtained,  by  merely  dividing  the  last  number  in  the  final 
column  of  the  work  by  the  last  number  in  the  previous 
column. 

The  method  by  which  we  obtained  the  first  four  figures 
of  the  root  in  the  last  example,  if  pursued  further,  would 
give  .000127  for  the  next  three  figures.  The  first  figure 
.0004  is  obtained,  according  to  the  rule  above,  by  dividing 
.002171875  by  5.076875.  But,  if  we  continue  the  division, 
we  shall  obtain  the  three  figures  .000427.  Now  to  show 
that  this  division  ought  to  give  two  or  three  figures  of  the 
root  correctly,  we  observe  that  the  equation  whose  co-effi- 
cients are  marked  4,  when  completed  by  putting  some  letter 
as  y  for  the  unknown  quantity,  is 

y3  +  6. 575?/2  +5. 076875?/-.  002171875=0. 
And  by  transposition, 

5. 076875?/=. 002171875-6. 575*/2-y3. 

Now,  y  represents  the  part  which  is  to  be  added  to  the 
number  3.525  already  found,  to  give  the  whole  root  of  the 
primitive  equation.  (Art.  574.)  And  the  value  of  y  is  less 
than  .0005;  for  its  first  figure,  obtained  by  dividing  .00217 
&3.  by  5.07  &?,.  is  .0004."  Therefore  the  term  6.575?/a  is 
less  than  .0000016  &c.  and  ?/3  is  much  less  than  this. 
Hence,  the  preceding  term  .002171975  can  not  be  affected, 


RESOLUTION      OF     ECIUATIONS.  373 

by  the  subtraction  of  the  other  two,  to  the  amount  of  more 
than  one  or  two  units  in  the  sixth  decimal  place.  If  then 
we  neglect  the  terms  containing  y2  and  */3,  and  divide 
.002171875  by  5.076875,  till  the  dividend,  as  far  as  the  sixth 
decimal  place,  is  exhausted,  the  quotient  .000427  will,  for 
so  many  figures,  be  the  true  value  of  y. 

A  greater  number  of  figures  of  the  root  might  have  been 
found  correctly  by  dividing  in  this  way,  if  a  greater  number 
had  been  obtained  by  the  previous  calculation. 

o76.  If  an  equation  has  two  or  more  real  roots,  they 
may  all  be  found  by  the  rule  above.  Or,  when  one  has  been 
obtained,  the  given  equation  may  be  reduced  (Art.  538,)  to 
an  equation  of  a  lower  degree,  and  another  root  deduced 
from  this. 

When  an  equation  has  been  depressed  to  a  quadratic,  the 
two  roots  of  this  may  be  found  by  the  rule  in  Art.  322.  We 
have  seen  above,  that  3.525427  is  one  of  the  roots  of  the 
equation 

#3  —  4x2  —  4#+20=0. 

If  we  divide  the  equation  by  x— 3.525427,  we  obtain  the 
quadratic 

x2  -  .474573^-5.6730716=0 ; 

whence  x  =. 2372865 drv/5. 7293757  ;  that  is,  x  equals 
2.630898,  or  —2.156325.     Hence  the  three  roots  of  the 

C      3.525427 
above  cubic  equation  are  <      2.630898 

(  -2.156325 

By  observing  that  the  quadratic  multiplied  by  x-3. 525427, 
ought  to  produce  the  cubic  equation,  we  shall  see  that  the 
term  -5.6730716,  may  be  found  by  simply  dividing  20  by 
—3.525427;  and  that  the  co-efficient  —.474573  is  equal 
to  —4,  the  co-efficient  of  the  second  term  in  the  cubic  equa- 
tion, diminished  by  —3.525427. 

577.  If  a  negative  root  is  required,  we  may  change  the 
signs  of  the  alternate  terms  in  the  proposed  equation,  and 
find  the  equal  positive  root.    (Art.  550.) 

Ex.  3.  Find  a  root  of  the  equation 

x3  —  5x+6=0. 

32 


374 


RESOLUTION      OF      EQUATIONS. 


This  equation  has  a  negative  root  between  —2  and  —3. 
(Art.  553.)  Putting  =b0  for  the  absent  term,  and  changing 
the  signs  of  the  alternate  terms,  we  have  the  new  equation 

z*±zO— 5:r— 6=0; 

which  has  a  positive  root,  equal  to  the  negative  root  of  the 
original  equation  (Art.  550.)  This  positive  root  is  found  as 
follows. 

1 


0 

-5 

-6  (2.689095 

2 

-1 

i-8 

4 

i  7 

a-1424 

i*6 

1096 

3-  151168 

72 

al528 

4—   1591231 

a  788 

159104 

s-     88870986571 

796 

3  165472 

c-      5402709467625 

s8049 

16619641 

8058 

4  16692163 

4  806709 

166928890381 

806718 

5  166936150843 

* 8067275 

16693655420675 

8067280 

6  16693695757075 

Now  by  dividing  the  last  number  in  the  final  column  by 
the  last  in  the  previous  column,  we  may  find  correctly  six  or 
seven  other  figures  of  the  root,  namely  3236377. 

Hence,  one  of  the  roots  of  the  original  equation  is 
-'2.6890953236377.  The  other  two  will  be  found  to  be 
imaginary. 

The  work  may  often,  as  in  this  example,  be  considerably 
abridged,  by  omitting  the  decimal  point  and  the  ciphers  at 
the  left  hand  of  the  decimal  numbers  in  the  final  column  or 
elsewhere.  But  care  must  be  taken  that  none  of  the  figures 
lose  their  proper  local  values. 

Ex.  4.    Find  a  root  of  the  equation 

9z3+70x2 +500^-547000=0. 

Having  found  by  trial  (Art.  553,)  that  the  equation  has  a 
root  between  30  and  40,  we  proceed  as  follows. 


RESOLUTION      OP 

EaU  ATIONS. 

9          70 

500 

-547000  (36.448 

340 

10700 

-226000 

610 

29000 

-18376 

880 

34604 

—  1995904 

934 

40532 

-339422144 

988 

4095024 

-7721193472 

1042 

4136992 

10456 

414120464 

10492 

414541872 

10528 

41462618816 

105316 

41471051008 

105352 

105388 

1053952 

• 

1054024 

375 


By  dividing  now,  as  in  previous  examples,  we  find  six  other 
figures  of  the  root,  namely  186182.     Therefore 

x=36. 448186182. 

Ex.  5.  Find  a  root  of  the  equation 

x3-48261145359159368=0; 
in  other  words,  extract  the  cube  root  of  48261145359159368. 

The  required  root  is  between  300000  and  400000;   and 
is  found  as  follows. 


0 

0 

-48261145359159368(364082 

3 

9 

-21261 

6 

27 

-1605145 

96 

3276 

-32601359159 

102 

3888 

-795329847368 

1084 

393136 

0 

1088 

397488 

109208 

3975753664 

109216 

3976627392 

1092242  397664923684 


Ans.  #=364082. 


Most  of  the  numbers  in  the  above  calculation  have  for  con- 
venience been  shortened  by  omitting  ciphers  at  the  right.  Thus, 
in  the  second  line,  3  stands  for  300000,  9  for  90000000000, 
and  21261  for  21261000000000000. 

We  may  observe  that  the  above  root  might  have  been  ob- 
tained, by  "first  finding  the  root  of  48.261145359159368,  and 
then  multiplying  the  result  by  100000. 


376 


RESOLUTION      OF      EQUATIONS, 


Ex.  6.  Find  a  root  of  the  equation 

z4 -fa:2 -8*- 15=0. 
Having  found  by  trial  that  2  is  the  first  figure  of  one  of 
the  roots,  we  proceed  as  follows. 


Ci 
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Ci     O     Ci     Ci     Ci     Ci     Ci 


It  will  be  seen,  by  examining  the  calculation  above,  that 
all  the  figures  at  the  right  of  the  vertical  lines,  and  all  in  the 
first  column  below  the  horizontal  line,  might  have  been  omit- 
ted, without  affecting  the  result.  The  work,  as  thus  abridged, 
is  exhibited  below. 


RESOLUTION      OF      EQUATIONS. 


377 


-15(2.302775 
i-ll 
2-1259 
s-35232966384 
4-3437544263 
5-256233658 
6-28984881 


45449919 


=  .6377323 


z=2. 3027756377323 


10             1  -8 

2              5  2 

4            13  i28 

6          i25  36247 

18            2749  245268 

83          3007  45333516808 

86        23274  34539907043 

89          32758404  4542203160 

292          3277681  4454449972              28984881 

9202    3327952  454472943 

920        328016  545449591 

92          32808  45449755 

39          43281  645449919 
9            3281 
328 
532 
32 
32 

In  the  former  case,  the  number  83  in  the  first  column  stood 
for  the  two  numbers  8  and  83,  and  the  number  9202  for  92 
and  9202.  (See  remark  under  Ex.  1.)  Here,  all  the  four 
numbers  are  distinctly  given. 

By  attending  to  the  above  operation,  it  will  be  seen  that, 
in  each  column  the  contraction  begins  in  the  second  number 
below  the  one  marked  2,  and  that  the  numbers  go  on  dimin- 
ishing in  extent  by  one  figure  on  the  right,  except  that  each 
of  the  numbers  marked  3,  4,  5.  &c.  extends  only  as  far  to  the 
right  as  the  one  below  it,  and  the  last  numbers  of  the  second 
and  third  columns  have  the  same  extent  as  those  above  them. 

If  we  had  made  the  calculation  at  first  in  the  abridged 
form  as  above,  the  final  figures  of  several  of  the  numbers 
would  have  been  different  from  those  we  have  obtained  by 
shortening  the  numbers  which  the  complete  calculation  had 
furnished.  Thus,  the  ninth  number  in  the  second  column, 
obtained  from  the  first  seven  figures  of  the  eighth  by  the  ad- 
dition of  2X920,  would  be  3277680  instead  of  3277681. 
And  the  seventh  number  in  the  third  column,  obtained  from 
the  one  above  it  by  omitting  the  final  figure  and  adding 
2X3277680.  would  have  0  for  its  last  figure,  instead  of  3.  But 
these  slight  differences  would  not  affect  more  than  one  or  two 
of  the  final  figures  in  the  value  of  x.  (See  remark  under  Ex.  7.) 
32* 


378 


RESOLUTION      OF     EttUATIONS, 


In  different  cases,  we  may  begin  to  abridge  the  work  at 
different  periods,  according  to  the  degree  of  accuracy  re- 
quired in  the  result. 

Ex.  7.  Find  a  root  of  the  equation 
x3  —  6x+2=0. 

Let  the  required  root  be  that  which  lies  between  2  and  c; 
We  then  proceed  as  follows. 

1 


0 

-6 

2(2.261802 

2 

-2      i- 

-2 

4 

l(>           2 

-552 

ifl 

724 

3-16824 

62 

a  852 

4-7494419 

64 

89196 

s   -  20986968 

a  66 

393228 

6  -  2292506 

666 

9329581 

672 

4  9336363 

3678 

934179004 

6781 

5  93472177 

2292506  nAnn^ 

6782 

9347231 

93-47245= -24526° 

4  6783 

69347245 

67838 

6784 

1 678 

6 

6 

Ans.  x=2. 261802245$ 

Here  we  begin  to  abridge  the  work,  by  reducing  the  num- 
ber next  to  67838  in  the  first  column  from  67846  to  6784. 
We  then  multiply  6784  by  8,  and  add  the  product  to 
93417900  in  the  second  column.  This  gives  93472172; 
which  might  be  put  for  the  next  number  in  the  same  column. 
But  it  will  be  more  accurate  to  add  to  this,  the  amount  which 
would  have  been  carried  for  tens  in  multiplying  67846  by  8, 
if  the  last  figure  had  not  been  cut  off.  Now  6  multiplied  by 
8  gives  48 ;  and  this  may  be  increased  to  52,  by  taking  ac- 
count of  the  last  figure  of  the  number  934179004.  If  then 
for  52,  we  carry  5,  we  obtain  the  number  93472177,  as 
given  above.  In  a  similar  manner  the  final  figures  of  other 
numbers  are  determined. 

Ex.  8.  Extract  the  fifth  root  of  123456789 ;  or,  find  a  root 
of  the  equation 

Xs  — 123456789=0. 


RESOLUTION     OF     EdUATFONS. 


379 


The  required  root  is  evidently  between  40  and  50 ;   and 
may  be  found  as  follows, 


0 

0 

0 

0      -123456789(41.52436 

4 

16 

64 

256       -21056789 

8 

48 

256 

1280        -7600588 

12 

96 

640 

13456201     -36176890625 

16 

160 

656201 

14128805      -6486836874 

20 

16201 

672604 

144776381875    -541957398 

201 

16403 

689210 

14830725312      -95999085 

202 

16606 

697666375 

14845026875      -6805875 

203 

16810 

70617425 

1485933533 

204 

1691275 

7147337 

1486219869 

205 

170157 

7150781 

148650623 

2055 

17119 

715423 

148652771   6805875 
14865492   14865578 — 45/yZ78 

206 

1722 

71577 

20 

1722 

71584 

14865535 

2 

172 

7159 

14865578 

17 

716 

1 

716 

1 

71 

7 
7 

Ans.  #=41. 524364578278. 

We  may  rely  upon  the  accuracy  of  all  but  one  or  two  of 
the  last  figures  in  this  value  of  x. 

Find  the  roots  of  the  following  equations. 

Ex.  9.  z3+5;c2+7£-47=0. 

Ans.  x=2. 1238314040872486. 
Ex.  10.     x*+3x2  +5.Z-1881718027578170. 433=0. 

Ans.  z=123456.7 
Ex.  11.  x3  -27^-36=0. 

(      5.765722 

Ans.   x=  ]  -4.320684 

f  -1.445038 

Ex.  12.      ar3-8169.5z2-8169. 5^-8170. 5=0. 

C      8170. 5_ 
Ans.  x=  1  -i+i^11^ 


380    APPLIC  AT  ION     OF     ALGEBRA     TO     GEOMETRY. 

x*  -484476471864=0. 

a;5 -f-x2 +x- 13099751099=0. 

x*+5x3+4x2+3x-105=0. 
Ans.  One  value  of  x  is  2.21733882735297. 

x*  —  &0x3  +  1998x2  —  14937:r+5000=0. 

Ans.  One  value  of  x  is   .3509870458. 

z4-19:c3  +  123.r2-302x+200=0. 

(  1.02804 

Ans    .-<4-00000 
Ans.   x-  ^6#57653 

V 7.39543 

Ex.  18.         x5  -7x*  -H5;c3  -58z2  +44^-300=0. 

Ans.  x=6. 119538. 


Ex. 

13. 

Ex. 

14. 

Ex. 

15. 

Ex. 

16. 

Ex. 

17. 

SECTION    XXI. 


APPLICATION    OF    ALGEBRA    TO    GEOMETRY.* 

Art.  57  S*  It  is  often  expedient  to  make  use  of  the  alge- 
braic notation  for  expressing  the  relations  of  geometrical 
quantities,  and  to  throw  the  several  steps  in  a  demonstration 
into  the  form  of  equations.  By  this,  the  nature  of  the  rea- 
soning is  not  altered.  It  is  only  translated  into  a  different 
language.  Signs  are  substituted  for  words,  but  they  are 
intended  to  convey  the  same  meaning.  A  great  part  of  the 
demonstrations  in  Euclid,  really  consist  of  a  series  of  equa- 
tions, though  they  may  not  be  presented  to  us  under  the 
algebraic  forms.  Thus  the  proposition,  that  the  sum  of  the 
three  angles  of  a  triangle  is  equal  to  two  right  angles,  (Euc. 
32,  1,)  may  be  demonstrated,  either  in  common  language,  or 
by  means  of  the  signs  used  in  Algebra. 

*  This  Section  is  to  be  read  after  the  Elements  of  Geometry. 


APPLICATION     OF     ALGEBRA     TO     GEOMETRY.    381 

Let  the  side  AB,  of  the  triangle  ABC,  (Fig.  1,)  be  con- 
tinued to  D ;  let  the  line  BE  be  parallel 
to  AC\  and  let  GHI  be  a  right  angle. 

The   demonstration,  in  words,   is  as 
follows  : 

1.  The  angle  EBD  is  equal  to  the  an- 

gle BAC3  (Euc.  29,  1.) 

2.  The  angle  CBE  is  egwaZ  to  the  an- 

gle ACB. 

3.  Therefore,  the  angle  EBD  added  to  CBE,  that  is,  the 

angle  CBD,  is  e^waZ  to  BAC  added  to  ACB. 

4.  If  to  these  equals,  we  add  the  angle  ABC,  the  angle  CBD 

added  to  ABC,  is  egwaZ  to  2?JlC  added  to  JLC2?  and 

5.  But  CBD  added  to  ABC,  is  e^waZ  to  twice  GHI,  that  is, 

to  two  right  angles.     (Euc.  13,  1.) 

6.  Therefore,  the  angles  BAC,  and  ACB,  and  ABC,  are 

together  equal  to  twice  GHZ",  or  two  right  angles. 

Now  by  substituting  the  sign  +,  for  the  word  added,  or 
and,  and  the  character  =,  for  the  word  equal,  we  shall  have 
the  same  demonstration  in  the  following  form. 

1.  By  Euclid  29,  1,  EBD=BAC 

2.  And  CBE=ACB 

3.  Add  the  two  equations  EBD+CBE=BAC+ACB 

4.  Add  ABC  t6  both  sides  CBD+ABC=BAC+ACB+ABC 

5.  But  by  Euclid  13,  1,      CBD+ABC=2GHI 

6.  Make  the  4th  &  5th  equal  BAC+ACB+ABC=2GHL 

By  comparing,  one  by  one,  the  steps  of  these  two  demon- 
strations, it  will  be  seen,  that  they  are  precisely  the  same, 
except  that  they  are  differently  expressed.  The  algebraic 
mode  has  often  the  advantage,  not  only  in  being  more  concise 
than  the  other,  but  in  exhibiting  the  order  of  the  quantities 
more  distinctly  to  the  eye.  Thus,  in  the  fourth  and  fifth  steps 
of  the  preceding  example,  as  the  parts  to  be  compared  are 
placed  one  under  the  other,  it  is  seen,  at  once,  what  must  be 
the  new  equation  derived  from  these  two.  This  regular  ar- 
rangement is  very  important  when  the  demonstration  of  a 
theorem,  or  the  resolution  of  a  problem,  is  unusually  compli- 
cated.    In  ordinary  language,  the  numerous  relations  of  the 


382    APPLICATION     OP     ALGEBRA     TO     GEOMETRY. 

quantities,  require  a  series  of  explanations  to  make  them  un- 
derstood ;  while  by  the  algebraic  notation,  the  whole  may  be 
placed  distinctly  before  us,  at  a  single  view.  The  disposition 
of  the  men  on  a  choee  board,  or  the  situation  of  the  objects 
in  a  landscape,  may  be  better  comprehended,  by  a  glance  of 
the  eye,  than  by  the  most  labored  description  in  words. 

579.  It  will  be  observed,  that  the  notation  in  the  exam- 
ple just  given  differs,  in  one  respect,  from  that  which  is  gene- 
rally used  in  algebra.  Each  quantity  is  represented,  not  by 
a  single  letter,  but  by  several.  In  common  algebra,  when 
one  letter  stands  immediately  before  another,  as  ab,  without 
any  character  between  them,  they  are  to  be  considered  as 
multiplied  together. 

But  in  geometry,  AB  is  an  expression  for  a  single  line, 
and  not  for  the  product  of  A  into  B.  Multiplication  is  de- 
noted, either  by  a  point  or  by  the  character  X.  The  pro- 
duct of  AB  into  CD,  is  AB.CD,  or  ABxCD. 

580.  There  is  no  impropriety,  however,  in  representing 
a  geometrical  quantity  by  a  single  letter.  We  may  make  b 
stand  for  a  line  or  an  angle,  as  well  as  for  a  number. 

If,  in  the  example  above,  we  put  the  angle 
EBD=a,  ACB^d,  ABC=h, 

BAC=b,  CBD=g,  GHI=l; 

CBE=c, 
the  demonstration  will  stand  thus ;  fc 

1.  By  Euclid  29,  1,  a=b 

2.  And  c=d 

3.  Adding  the  two  equations,  a-\-c=g=b-\-d 

4.  Adding  h  to  both  sides,  g-\-h=b+d+h 

5.  By  Euclid  13,  1,  g+h=2l 

6.  Making  the  4th  and  5th  equal,  b+d+h=2l. 

This  notation  is,  apparently,  more  simple  than  the  other; 
but  it  deprives  us  of  what  is  of  great  importance  in  geomet- 
rical demonstrations,  a  continual  and  easy  reference  to  the 
figure.  To  distinguish  the  two  methods,  capitals  are  gene- 
rally used,  for  that  which  is  peculiar  to  geometry ;  and  small 
letters,  for  that  which  is  properly  algebraic.  The  latter  has 
the  advantage  in  long  and  complicated  processes,  but  the 
other  is  often  to  be  preferred,  on  account  of  the  facility  with 
which  the  figures  are  consulted. 


1> 


APPLICATION     OF     ALGEBRA     TO     GEOMETRY.       3$3 

581.  If  a  line,  whose  length  is  measured  from  a  given 
point  or  line,  be  considered  positive ;  a  line  proceeding  in  the 
opposite  direction   is  to  be   considered  Fi„  2# 
negative.    If  AB  (Fig.  2,)  reckoned  from 
DE  on  the  right,  is  positive;  AC  on 
the  left  is  negative. 

A  line  may  be  conceived  to  be  pro-    H 

duced  by  the  motion  of  a  point.  Sup- 
pose a  point  to  move  in  the  direction  of 
AB,  and  to  describe  a  line  varying  in 
length  with  the  distance  of  the  point 
from  A.  While  the  point  is  moving  towards  B,  its  distance 
from  A  will  increase.  But  if  it  move  from  B  towards  C,  its 
distance  from  A  will  diminish,  till  it  is  reduced  to  nothing, 
and  then  will  increase  on  the  opposite  side.  As  that  which 
increases  the  distance  on  the  right,  diminishes  it  on  the  left, 
the  one  is  considered  positive,  and  the  other  negative.  See 
Arts.  54,  55. 

Hence,  if  in  the  course  of  a  calculation,  the  algebraic 
value  of  a  line  is  found  to  be  negative ;  it  must  be  measured 
in  a  direction  opposite  to  that  which,  in  the  same  process, 
has  been  considered  positive. 

582.  In  algebraic  calculations,  there  is  frequent  occasion 
for  multiplication,  division,  involution,  &c.  But  how,  it  may 
be  asked,  can  geometrical  quantities  be  multiplied  into  each 
other?  One  of  the  factors,  in  multiplication,  is  always  to  be 
considered  as  a  number.  (Art.  88.)  The  operation  consists  in 
repeating  the  multiplicand  as  many  times  as  there  are  units 
in  the  multiplier.  How  then  can  a  line,  a  surface,  or  a  solid, 
become  a  multiplier? 

To  explain  this,  it  will  be  necessary  to  observe,  that  when- 
ever one  geometrical  quantity  is  multiplied  into  another, 
some  particular  extent  is  to  be  considered  the  unit.  It  is 
immaterial  what  this  extent  is,  provided  it  remains  the  same, 
in  different  parts  of  the  same  calculation.  It  may  be  an 
inch,  a  foot,  a  rod,  or  a  mile.  If  an  inch  is  taken  for  the 
unit,  each  of  the  lines  to  be  multiplied,  is  to  be  considered  as 
made  up  of  so  many  parts,  as  it  contains  inches.  The  mul- 
tiplicand will  then  be  repeated,  as  many  times,  as  there  are 
units  in  the  multiplier.  If,  for  instance,  one  of  the  lines  be  a 
foot  long,  and  the  other  half  a  foot ;  the  factors  will  be,  one 
12  inches,  and  the  other  6,  and  the  product  will  be  72  inches. 
Though  it  would  be  absurd  to  say  that  one  line  is  to  be  re- 


384    APPLICATION     OF     ALGEBRA     TO     GEOMETRY. 


peated  as  often  as  another  is  long;  yet  there  is  no  impro- 
priety in  saying,  that  one  is  to  be  repeated  as  many  times,  as 
there  are  feet  or  rods  in  the  other.  This,  the  nature  of  a 
calculation  often  requires. 

•5851.  If  the  line  which  is  to  be  the  multiplier,  is  only  a, 
part  of  the  length  taken  for  the  unit;  the  product  is  a  like 
part  of  the  multiplicand.  (Art.  85.)  Thus,  if  one  of  the 
factors  is  6  inches,  and  the  other  half  an  inch,  the  product 
is  3  inches. 

•584:.  Instead  of  referring  to  the  measures  in  common  use, 
as  inches,  feet,  &c.  it  is  often  convenient  to  fix  upon  one  of 
the  lines  in  a  figure,  as  the  unit  with  which  to  compare  all  the 
others.  When  there  are  a  number  of  lines  drawn  within  and 
about  a  circle,  the  radius  is  commonly  taken  for  the  unit. 
This  is  particularly  the  case  in  trigonometrical  calculations. 

•585.  The  observations  which  have  been  made  concern- 
ing lines,  maybe  applied  to  surfaces  and  solids.  There  may 
be  occasion  to  multiply  the  area  of  a  figure,  by  the  number 
of  inches  in  some  given  line. 

But  here  another  difficulty  presents  itself.  The  product 
of  two  lines  is  often  spoken  of,  as  being  equal  to  a  surface; 
and  the  product  of  a  line  and  a  surface,  as  equal  to  a  solid. 
Thus  the  area  of  a  parallelogram  is  said  to  be  equal  to  the 
product  of  its  base  and  height ;  and  the  solid  contents  of  a 
cylinder,  are  said  to  be  equal  to  the  product  of  its  length  into 
the  area  of  one  of  its  ends.  But  if  a  line  has  no  breadth, 
how  can  the  multiplication,  that  is  the  repetition,  of  a  line 
produce  a  surface  ?  And  if  a  surface  has  no  thickness,  how 
can  a  repetition  of  it  produce  a  solid  ? 

If  a  parallelogram,  represented  on  a  reduced  scale  by 
ABCD,  (Fig.  3,)  be  five  inches  long, 
and  three  inches  wide  ;  the  area  or  sur- 
face is  said  to  be  equal  to  the  product 
of  5  into  3,  that  is,  to  the  number  of 
inches  in  AB,  multiplied  by  the  num- 
ber in  EC.  But  the  inches  in  the  lines 
AB  and  BC  are  linear  inches,  that  is, 
inches  in  length  only ;  while  those 
which  compose  the  surface  AC  are 
superficial  or  square  inches,  a  different 
species  of  magnitude.     How  can  one  of  these  be  converted 


Fig.  3. 


9 
A  B 

o  i      I  pi 


APPLICATION     OF     ALGEBRA     TO     GEOMETRY.    385 

into  the  other  by  multiplication,  a  process  which  consists  in 
repeating  quantities,  without  changing  their  nature? 

586.  In  answering  these  inquiries,  it  must  be  admitted, 
that  measures  of  length  do  not  belong  to  the  same  class  of 
magnitudes  with  superficial  or  solid  measures  ;  and  that  none 
of  the  steps  of  a  calculat'on  can,  properly  speaking,  trans- 
form the  one  into  the  other.  But,  though  a  line  can  not 
become  a  surface  or  a  solid,  yet  the  several  measuring  units 
in  common  use  are  so  adapted  to  each  other,  that  squares, 
cubes,  &c.  are  bounded  by  lines  of  the  same  name.  Thus 
the  side  of  a  square  inch,  is  a  linear  inch  ;  that  of  a  square 
rod,  a  linear  rod,  &c.  The  length  of  a  linear  inch  is,  there- 
fore, the  same  as  the  length  or  breadth  of  a  square  inch. 

If  then  several  square  inches  are  placed  together,  as  from 
Q  to  R,  (Fig.  3,)  the  number  of  them  in  the  parallelogram 
OR  is  the  same  as  the  number  of  linear  inches  in  the  side 
QR  :  and  if  we  know  the  length  of  this,  we  have  of  course 
the  area  of  the  parallelogram,  which  is  here  supposed  to  be 
one  inch  wide. 

But,  if  the  breadth  is  several  inches,  the  larger  parallelo- 
gram contains  as  many  smaller  ones,  each  an  inch  wide,  as 
there  are  inches  in  the  whole  breadth.  Thus,  if  the  paral- 
lelogram AC  (Fig.  3,)  is  5  inches  long,  and  3  inches  broad, 
it  may  be  divided  into  three  such  parallelograms  as  OR.  To 
obtain,  then,  the  number  of  squares  in  the  large  parallelo- 
gram, we  have  only  to  multiply  the  number  of  squares  in  one 
of  the  small  parallelograms,  into  the  number  of  such  paral- 
lelograms contained  in  the  whole  figure.  But  the  number  of 
square  inches  in  one  of  the  small  parallelograms  is  equal  to 
the  number  of  linear  inches  in  the  length  AB.  And  the 
number  of  small  parallelograms,  is  equal  to  the  number  of 
linear  inches  in  the  breadth  BC.  It  is  therefore  said  con- 
cisely, that  the  area  of  the  parallelogram  is  equal  to  the 
length  multiplied  into  the  breadth. 

587.  We  hence  obtain  a  convenient  algebraic  expression, 
for  the  area  of  a  right-angled  parallelogram.  If  two  of  the 
sides  perpendicular  to  each  other  are  AB  and  BC,  the  expres- 
sion for  the  area  is  ABxBC ;  that  is,  putting  a  for  the  area, 

a=ABxBC. 
It  must  be  understood,  however,  that  when  AB  stands  for 
a  line,  it  contains  only  linear  measuring  units;  but  when  it 
enters  into  the  expression  for  the  area,  it  is  supposed  to  con- 

33 


386    APPLICATION      OF     ALGEBRA     TO     GEOMETRY. 


Fig.  4. 

A 

B 

Ii 

a 

i 

/ 


tain  superficial  units  of  the  same  name.  Yet  as,  in  a  given 
length,  the  number  of  one  is  equal  to  that  of  the  other,  they 
may  be  represented  by  the  same  letters,  without  leading  to 
error  in  calculation. 

588.  The  expression  for  the  area  may  be  derived,  by  a 
method  more  simple,  but  less  satisfac- 
tory perhaps  to  some,  from  the  princi- 
ples which  have  been  stated  concerning 
variable  quantities,  in  the  12th  section. 
Let  a  (Fig.  4,)  represent  a  square  inch, 
foot,  rod,  or  other  measuring  unit ;  and 
let  b  and  /  be  two  of  its  sides.  Also, 
let  A  be  the  area  of  any  right-angled 
parallelogram,  B  its  breadth,  and  L  its 
length.  Then  it  is  evident,  that,  if  the 
breadth  of  each  were  the  same,  the  areas  would  be  as  the 
lengths ;  and,  if  the  length  of  each  were  the  same,  the  areas 
wouid  be  as  the  breadths. 

That  is,         A  :  a  : :  L  :  /,  when  the  breadth  is  given ; 
And  A:  a:  :  B:b,  when  the  length  is  given  ; 

Therefore,  (Art.  430,)  A:a::BxL:bl,  when  both  vary. 
That  is,  the  area  is  as  the  product  of  the  length  and  breadth, 

589.  Hence,  in  quoting  the  Elements  of  Euclid,  the 
term  product  is  frequently  substituted  for  rectangle.  And 
whatever  is  there  proved  concerning  the  equality  of  certain 
rectangles,  may  be  applied  to  the  product  of  the  lines  which 
contain  the  rectangles.* 

590.  The  area  of  an  oblique  parallelogram  is  also  ob- 
tained by  multiplying  the  base  into  the 
perpendicular  height.  Thus  the  ex- 
pression for  the  area  of  the  parallelo- 
gram ABNM  (Fig.  5,)  is  MNxAD  or 
ABxBC.  For  by  Art.  587,  ABxBC 
is  the  area  of  the  right-angled  parallelo- 
gram ABCD,  and  by  Euclid  36,  l,f 

Earallelograms  upon  equal  bases,  and 
etween  the  same  parallels,  are  equal;   that  is,  ABCD  is 
equal  to  ABNM. 


Fig.  5. 


*  See  Note  Q. 

|  Legendre's  Geometry,  American  Edition,  Art.  166. 


APPLICATION     OF     ALGEBRA     TO     GEOMETRY.    387 


•591.  The  area  of  a  square  is  obtained,  by  multiplying 
me  of  the  sides  into  itself.     Thus  the  ex- 
area  of  the  square  AC, 


Fig.  6, 


that  is, 


is   equal   to   ABxBC. 


session  for  the 
Tig.  6,)  is  AB\ 

a=AB. 

For   the    area 
[Art.  587.) 
But  AB—BC,  therefore, 

ABxBC=ABxAB=JLB\ 

592.  The  area  of  a  triangle  is  equal  to  half  the  product 
of  the  base  and  height.     Thus  the  area  p.    ^ 

of  the  triangle  ABG,  (Fig.  7,)  is  equal 
to  half  AB  into  GH  or  its  equal  BC> 
that  is,  a=\ABxBC. 

For  the  area  of  the  parallelogram 
ABCD  is  ABxBC,  (Art.  587.)  And 
by  Euclid  41,  1,#  if  a  parallelogram  and 
a  triangle  are  upon  the  same  base,  and 
between  the  same  parallels,  the  triangle  is  half  the  parallelo- 
gram. 

593.  Hence,  an  algebraic  expres- 
sion may  be  obtained  for  the  area  of 
any  figure  whatever,  which  is  bounded 
by  right  lines.  For  every  such  figure 
may  be  divided  into  triangles.  a 

Thus  the  right-lined  figure 
ABCDE    (Fig.  8,)    is    composed   of 
the  triandes  ABC,  ACE,  and  ECD. 


Fig.  8. 


The  area  of  the  triangle 
That  of  the  triangle 


ABC=\ACxBLy 

ACE=\ACxEHt 

ECD=iECxDG. 

therefore,  equal  to 


That  of  the  triangle 

The  area  of  the  whole  figure  is, 

QACxBL)  +  QACxEH)  +  (lECxDG). 

The  explanations  in  the  preceding  articles  contain  the  first 
principles  of  the  mensuration  of  superficies.     The  object  of 


*  Legendre,  168. 


388    APPLICATION     OF     ALGEBRA     TO     GEOMETRY. 

introducing  the  subject  in  this  place,  however,  is  not  to  make 
a  practical  application  of  it,  at  present;  but  merely  to  show 
the  grounds  of  the  method  of  representing  geometrical  quan- 
tities in  algebraic  language. 

•59 1.  The  expression  for  the  superficies  has  here  been 
derived  from  that  of  a  line  or  lines.  It  is  frequently  neces- 
sary to  reverse  this  order;  to  find  a  side  of  a  figure,  from 
knowing  its  area. 

If  the  number  of  square  inches  in  the  parallelogram 
ABCD  (Fig.  3,)  whose  breadth  BC  is  3  inches,  be  divided 
by  3 ;  the  quotient  will  be  a  parallelogram  ABFE,  one  inch 
wide,  and  of  the  same  length  with  the  larger  one.  But  the 
length  of  the  small  parallelogram,  is  the  length  of  its  side 
AB.  The  number  of  square  inches  in  one  is  the  same,  as 
the  number  of  linear  inches  in  the  other.  (Art.  586.)  If 
therefore,  the  area  of  the  large  parallelogram  be  represented 

by  a,  the  side  AB=  r—,  that  is,  the  length  of  a  parallelo- 
gram is  found  by  dividing  the  area  by  the  breadth. 

595.  If  a  be  put  for  the  area  of  a  square  whose  side  is 
AB} 

Then  by  Art.  591,  a^AB 

And  extracting  both  sides,  v/a= AB 

That  is,  the  side  of  the  square  is  found,  by  extracting  the 
square  root  of  the  number  of  measuring  units  in  its  area. 

090.  If  AB  be  the  base  of  a  triangle  and  BC  its  per- 
pendicular height; 

Then  by  Art.  592,  a=%BCxAB 

And  dividing  by  \BC,  1  Rr,=AB. 

That  is,  the  base  of  a  triangle  is  found,  by  dividing  the 
area  by  half  the  height 

59 7o  As  a  surface  is  expressed,  by  the  product  of  its 
length  and  breadth  ;  the  contents  of  a  solid  may  be  expressed, 
by  the  product  of  its  length,  breadth  and  depth.  It  is  neces- 
sary to  bear  in  mind,  that  the  measuring  unit  of  solids,  is  a 
cube;  and  that  the  side  of  a  cubic  inch,  is  a  square  inch; 
the  side  of  a  cubic  foot,  a  square  foot,  &c. 


APPLICATION     OF     ALGEBRA     TO     GEOMETRY.    389 

Let  ABCD  (Fig.  3,)  represent  the  base  of  a  parallelopiped, 
five  inches  long,  three  inches  broad,  and  one  inch  deep.  It 
is  evident  there  must  be  as  many  cubic  inches  in  the  solid,  as 
there  are  square  inches  in  its  base.  And,  as  the  product  of 
the  lines  AB  and  BC  gives  the  area  of  this  base,  it  gives,  of 
course,  the  contents  of  the  solid.  But  suppose  that  the 
depth  of  the  parallelopiped,  instead  of  being  one  inch,  is  four 
inches.  Its  contents  must  be  four  times  as  great.  If,  then, 
the  length  be  AB,  the  breadth  BC,  and  the  depth  CO,  the 
expression  for  the  solid  contents  will  be, 
ABxBCxCO. 

«598.  By  means  of  the  algebraic  notation,  a  geometrical 
demonstration  may  often  be  rendered  much  more  simple  and 
concise, than  in  ordinary  language.  The  proposition,  (Eua 
4,  2,)  that  when  a  straight  line  is  divided  into  two  parts,  the 
square  of  the  whole  line  is  equal  to  the  squares  of  the  two 
parts,  together  with  twice  the  product  of  the  parts,  is  demon- 
strated, by  involving  a  binomial. 

Let  the  side  of  a  square  be  represented  by  s ; 

And  let  it  be  divided  into  two  parts,  a  and  b. 

By  the  supposition,  s  =a   -\-b 

And  squaring  both  sides,  s2—a2  +2ab  +  b2. 

That  is,  s2  the  square  of  the  whole  line,  is  equal  to  a2 
and  b2,  the  squares  of  the  two  parts,  together  with  2ab, 
twice  the  product  of  the  parts. 

•599.  The  algebraic  notation  may  also  be  applied,  with 
great  advantage,  to  the  solution  of  geometrical  problems.  In 
doing  this,  it  will  be  necessary,  in  the  first  place,  to  raise  an 
algebraic  equation,  from  the  geometrical  relations  of  the 
quantities  given  and  required  ;  and  then  by  the  usual  reduc- 
tions, to  find  the  value  of  the  unknown  quantity  in  this  equa- 
tion.    See  Art.  195. 

Prob.  1.  Given  the  base,  and  the  sum  of  the  hypothenuse 
and  perpendicular,  of  the  right  angled  Fig>  9# 

triangle,  ABC,  (Fig.  9,)  to  find  the  per- 
pendicular. 

Let  the  base  AB=b 

The  perpendicular  BC=x 

The  sum  of  hyp.  &per.  x+AC=a 
Then  transposing  x,  AC—a-x- 

33*  A 


390    APPLICATION      OP     ALGEBRA     TO     GEOMETRY. 


1.  By  Euclid  47,  1,*  BC  +AB  =  AC 

2.  That  is,  by  the  notation,  x2  +b2  =  (a-x)2  =a2-2ax+x2 . 

Here  we  have  a  common  algebraic  equation,  containing 
only  one  unknown  quantity.     The  reduction  of  this  equation 
in  the  usual  manner,  will  give 
a2-b2 


2a 


=BC,  the  side  required. 


The  solution,  in  letters,  will  be  the  same  for  any  right 
angled  triangle  whatever,  and  may  be  expressed  in  a  general 
theorem,  thus;  'In  a  right  angled  triangle,  the  perpendicular 
is  equal  to  the  square  of  the  sum  of  the  hypothenuse  and 
perpendicular,  diminished  by  the  square  of  the  base,  and  divi- 
ded by  twice  the  sum  of  the  hypothenuse  and  perpendicular.' 

It  is  applied  to  particular  cases  by  substituting  numbers, 

for  the  letters  a  and  b.     Thus  if  the  base  is  8  feet,  and  the 

sum  of  the  hypothenuse  and  perpendicular  16,  the  expression 

a2—b2  162— 82 

— becomes  — — -—=6,    the  perpendicular;    and  this 

Zid  *X  lb 

subtracted  from  16,  the  sum  of  the  hypothenuse  and  perpen- 
dicular, leaves  10,  the  length  of  the  hypothenuse. 

Prob.  2.  Given  the  base  and  the  difference  of  the  hypothe- 
nuse and  perpendicular,  of  a  right  angled  triangle,  to  find  the 
perpendicular. 

Let  the  base      AB  (Fig.  10,)=&=20^ 
The  perpendicular, 
The  given  difference, 
Then  will  the  hypoth. 

Then 

1.  By  Euclid  47,  1, 

2.  That  is,  by  the  notation, 

3.  Expanding  (x-\-d)*t 

4.  Therefore, 


Fig.  10. 


AC  =AB  +BC 

(x+d)2=b2+x2 
x2+2dx  +  d2  =  b*+x* 
b2-d2 


27==- 


2d 


=  15. 


Prob.  3.   If  the  hypothenuse  of  a  right  angled  triangle  is 
30  feet,  and  the  difference  of  the  other  two  sides  6  feet,  what 

Ans.  24  feet. 


is  the  length  of  the  base  ? 


*  Legeudre,  186* 


APPLICATION      OF     ALGEBRA     TO     GEOMETRY.    391 


Prob.  4.  If  the  hypothenuse  of  a  right  angled  triangle  is 
50  rods,  and  the  base  is  to  the  perpendicular  as  4  to  3,  what 
is  the  length  of  the  perpendicular  ?  Ans.  30. 

Prob.  5.  Having  the  perimeter  and  the  diagonal  of  a  par- 
allelogram ABCD,  (Fig.  11,)  to  find  the  sides. 

Let  the  diagonal  AC=h=\0- 

The  side  AB=x 

Half  the  perimeter      BC+AB=BC+x=b=U i 
Then  by  transposing  x,  BC=b—x 

„.      . ,  Fig.  12. 

Fig.  11.  8 

D  C 


By  Euclid  47,  1, 
That  is, 

Therefore, 


AB  +BC=AC 

x2  +  (b-x)2=h2 


Here  the  side  AB  is  found ;  and  the  side  BC  is  .equal  to 
6-^=14-8=6. 

Prob.  6.  The  area  of  a  right  angled  triangle  ABC  (Fig.  12,) 
being  given,  and  the  sides  of  a  parallelogram  inscribed  in  it, 
to  find  the  side  BC. 


Let  the  given  area  =a, 

EB=DF=d, 
Then  by  the  figure, 

1.  By  similar  triangles, 

2.  That  is, 

3.  Therefore, 

4.  By  Art.  592, 

5.  Dividing  by  \x, 

6.  Therefore, 

7.  And 


DE=BF=b        ) 
BC=x        [ 
CF=BC-BF=x-b.  3 
CF:DF::BC:AB 

x—b  :  d: :  x  :  AB 
dx=(x-b)xAB 
a=ABx\BC=ABx\x 


2a 


=AB 


ix     2a  2a& 

dx—(x— b)X  —  =2a 

v         -      x  x 


a       \   /a2      2ab      _-ll 


392    APPLICATION     OF     ALGEBRA     TO     GEOMETRF, 

Prob.  7.   The  three  sides  of  a  right  £  Fie- 13- 

angled  triangle,  ABC,  (Fig.  13,)  being 
given,  to  find  the  segments  made  by 
a  perpendicular,  drawn  from  the  right 
angle  to  the  hypothenuse. 

The  perpendicular  will  divide  the 
original  triangle,  into  two  right  angled 
triangles,  BCD  and  ABD.    (Euc.  8,  6.) 


1.  By  Euc.  47,  1, 

2.  By  the  figure, 

3.  Squaring  both  sides,    CD~  =  (AC— AD)2 


BD  +CD  =BC 
CD=AC-AD 


4.  Therefore, 

5.  Expanding, 

6.  Transposing, 

7.  By  Euc.  47,  1, 


W  +  (AC-AD)  =BC 

BD2+ AC  -2 AC.  AD -f- AD* ]-=~BC 

bd2=bc2-ac2+2ac.  ad-ad* 
bd2=ab2-ad2 


8.  Making  6th  &  7th  eq.  BC°~  -AC2  +2AC .AV^AB* 


9.  Therefore, 


AD; 


AB~  +  AC  -BC 
2AC 


The  unknown  lines,  to  distinguish  them  from  those  which 
are  known,  are  here  expressed  by  Roman  letters. 

Prob.  8.  Having  the  area  of  a  parallelogram  DEFG  (Fig. 
14,)  inscribed  in  a  given  triangle,  ABC,  to  find  the  sides  of 
the  parallelogram. 

Draw  CI  perpendicular  to  AB.     By 
supposition,  DG  is  parallel  to  AB. 
Therefore, 

The  triangle  CHG,  is  similar  to  CIB ) 

And  CDG,  to  CAB  J 

Let  CI=d  DG=x  ) 

AB=b  The  given  area  =a  ) 

1.  By  similar  triangles,  CB  :  CG  : 

2.  And  CB.CG 

3.  By  equal  ratios,  (Art.  392,)  AB  :  Z>£  : 


AB .  DG 

:  C/:  CH 

CI:  CH 


APPLICATION     OF     ALGEBRA     TO     GEOMETRY.    393 


4.  Therefore, 

5.  By  the  figure, 

6.  Substituting  for  CH, 

7.  That  is 

8.  By  Art.  587, 

9.  That  is, 

10.  This  reduced  gives 


DGxCI 

AB 
CI-CH=IH 
DGxCI 


CI- 


=  CH 

BE 
=BE 


d- 


dx 


AB 
-DE 


a=BGxBE 

dx2 

~b~ 

b 


--xx(d-j) 


a=dx- 


/lb2      ab\ 


-DG. 


The  side  DE  is  found,  by  dividing  the  area  by  DG. 

Prob.  9.  Through  a  given  point,  in  a  given  circle,  so  to 
draw  a  right  line,  that  its  parts,  be-  Fj    lg 

tween   the  point  and   the  periphery, 
shall  have  a  given  difference. 

In  the  circle  AQBR,  (Fig.  15,)  let 
P  be  a  given  point  in  the  diameter 
AB. 

Let  AP=a,  PR=z, 

BP=by  The  given  difference  =d. 

Then  will      PQ=x+d. 


1.  By  Euc.  35,  3,* 

2.  That  is, 

3.  Or, 

4.  Completing  the  square, 

5.  Extracting  and  transposing, 


PRxPQ=APxBP 

xX{x+d)=aXb 

x2  -\-dx=ab 

x2  +dz+%d*  =\d2  +ab 

x=—iddc^ia-  +ab  =  PR. 


With  a  little  practice,  the  learner  may  very  much  abridge 
these  solutions,  and  others  of  a  similar  nature,  by  reducing 
several  steps  to  one. 

Prob.  10.  If  the  sum  of  two  of  the  sides  of  a  triangle  be 
1155,  the  length  of  a  perpendicular  drawn  from  the  angle 
included  between   these  to  the  third   side  be  300,  and  the 


*  Legendre,  224. 


394    APPLICATION     OF     ALGEBRA     TO     GEOMETRY. 

difference  of  the  segments  made  by  the  perpendicular,   be 
495 ;  what  are  the  lengths  of  the  three  sides  ? 

Ans.  945,  375,  and  780. 

Prob.  11.  If  the  perimeter  of  a  right  angled  triangle  be 
720,  and  the  perpendicular  falling  from  the  right  angle  on  the 
hypothenuse  be  144 ;  what  are  the  lengths  of  the  sides  ? 

Ans.  300,  240,  and  180. 

Prob.  12.  The  difference  between  the  diagonal  of  a  square 
and  one  of  its  sides  being  given,  to  find  the  length  of  the  sides. 
If  x=  the  side  required,  and  d=  the  given  difference ; 
Then     x=d+d^2. 

Prob.  13.  The  base  and  perpendicular  height  of  any  plane 
triangle  being  given,  to  find  the  side  of  a  square  inscribed  in 
the  triangle,  and  standing  on  the  base,  in  the  same  manner 
as  the  parallelogram  DEFG,  on  the  base  AB,  (Fig.  14.) 

If  x=  a  side  of  the  square,    b=  the  base,  and  A=  the 

height  of  the  triangle ; 

oh 
Then     x=r-— r- 
b+n 

Prob.  14.  Two  sides  of  a  triangle,  and  a  line  bisecting  the 
included  angle  being  given ;  to  find  the  length  of  the  base  or 
third  side,  upon  which  the  bisecting  line  falls. 

If  x—  the  base,  a=  one  of  the  given  sides,  c=  the  other, 
and  b=  the  bisecting  line ; 

Then     x=(a+c)X\/ 

v        '       v        ac. 

Prob.  15.  If  the  hypothenuse  of  a  right  angled  triangle  be 
35,  and  the  side  of  a  square  inscribed  in  it,  in  the  same  man- 
ner as  the  parallelogram  BEDF,  (Fig.  12,)  be  12 ;  what  ai'e 
the  lengths  of  the  other  two  sides  of  the  triangle? 

Ans.  28,  and  21. 

Prob.  16.  The  number  of  feet  in  the  perimeter  of  a  right 
angled  triangle,  is  equal  to  the  number  of  square  feet  in  the 
area ;  and  the  base  is  to  the  perpendicular  as  4  to  3.  Re- 
quired the  length  of  each  of  the  sides. 

Ans.  6,  8,  and  10. 

Prob.  17.  A  grass  plat  12  rods  by  18,  is  surrounded  by  a 
gravel  walk  of  uniform  breadth,  whose  area  is  equal  to  that 
of  the  grass  plat.     What  is  the  breadth  of  the  gravel  walk  ? 


APPLICATION     OF     ALGEBRA     TO     GEOMETRY.    395 

Prob.  18.  The  sides  of  a  rectangular  field  are  in  the  ratio 
of  6  to  5 ;  and  one-sixth  of  the  area  is  125  square  rods. 
What  are  the  lengths  of  the  sides  ? 

Prob.  19.  There  is  a  right  angled  triangle,  the  area  of 
which  is  to  the  area  of  a  given  parallelogram  as  5  to  8.  The 
shorter  side  of  each  is  60  rods,  and  the  other  side  of  the  tri- 
angle adjacent  to  the  right  angle,  is  equal  to  the  diagonal  of 
the  parallelogram.     Required  the  area  of  each  ? 

Ans.  4800  and  3000  square  rods. 

Prob.  20.  There  are  two  rectangular  vats,  the  greater  of 
which  contains  20  cubic  feet  more  than  the  other.  Their 
capacities  are  in  the  ratio  of  4  to  5 ;  and  their  bases  are 
squares,  a  side  of  each  of  which  is  equal  to  the  depth  of  the 
other  vat.     Required  the  depth  of  each  ? 

Ans.  4  and  5  feet. 

Prob.  21.  Given  the  lengths  of  three  perpendiculars,  drawn 
from  a  certain  point  in  an  equilateral  triangle,  to  the  three 
sides ;  to  find  the  length  of  the  sides. 

If  a,  b,  and  c,  be  the  three  perpendiculars,  and  x=  half 

the  length  of  one  of  the  sides ; 

a+b+c 
I  hen     x= — —  • 

Prob.  22.  A  square  public  green  is  surrounded  by  a  street 
of  uniform  breadth.  The  side  of  the  square  is  3  rods  less 
than  9  times  the  breadth  of  the  street ;  and  the  number  of 
square  rods  in  the  street,  exceeds  the  number  of  rods  in  the 
perimeter  of  the  square  by  228.  What  is  the  area  of  the 
square  ?  Ans.   576  rods. 

Prob.  23.  Given  the  lengths  of  twoiines  drawn  from  the 
acute  angles  of  a  right  angled  triangle,  to  the  middle  of  the 
opposite  sides ;  to  find  the  lengths  of  the  sides. 

If  x—  half  the  base,  y=  half  the  perpendicular,  and  a 
and  b  equal  the  two  given  lines ; 

/4&2-a3  /4a* -b*   # 

Then     x=s/-^~  y=v/__.* 

*  See  Note  R. 


NOTES. 


Note  A.    Page  1. 

As  the  term  quantity  is  here  used  to  signify  whatever  is 
the  object  of  mathematical  inquiry,  it  will  be  obvious  that 
number  is  meant  to  be  included  ;  so  far  at  least,  as  it  can  be 
the  subject  of  mathematical  invescigation.  Dugald  Stewart 
asserts,  indeed,  that  it  might  be  easily  shown,  that  number 
does  not  fall  under  the  definition  of  quantity  in  any  sense  of 
that  word.*  For  proof  that  it  is  included  in  the  common 
acceptation  of  the  word,  it  will  be  sufficient  to  refer  to  almost 
any  mathematical  work  in  which  the  term  quantity  is  ex- 
plained, and  particularly  to  the  familiar  distinction  between 
continued  quantity  or  magnitude,  and  discrete  quantity  or 
number. 

But  does  number  "fall  under  the  definition  of  quantity T 
Mr.  Stewart  after  quoting  the  observation  of  Dr.  Reid,  that 
the  object  of  the  mathematics  is  commonly  said  to  be  quan- 
tity, which  ought  to  be  defined,  that  which  may  be  measured, 
adds,  "  The  appropriate  objects  of  this  science  are  such 
things  alone  as  admit  not  only  of  being  increased  and  dimin- 
ished, but  of  being  multiplied  and  divided.  In  other  words, 
the  common  character  which  characterizes  all  of  them,  is 
their  mensur -ability :"  That  number  may  be  multiplied  and 
divided,  will  not  probably  be  questioned.  But  it  may  per- 
haps be  doubted,  whether  it  is  capable  of  mensuration.  If, 
as  Mr.  Locke  observes,  "  number  is  that  which  the  mind 
makes  use  of,  in  measuring  all  things  that  are  measurable," 
can  it  measure  itself,  or  be  measured?  It  is  evident  that  it 
can  not  be  measured  geometrically,  by  applying  to  it  a  meas- 
ure of  length  or  capacity.  But  by  measuring  a  quantity 
mathematically,  what  else  is  meant,  than  determining  the 
ratio  which  it  bears  to  some  other  quantity  of  the  same  kind ; 
in  other  words  finding  how  often  one  is  contained  in  the 
other,  either  exactly  or  with  a  certain  excess?  And  is  not 
this  as  applicable  to  number  as  to  magnitude?     The  ratio 

*  Philosophy  of  the  Mind,  Vol.  II,  Note  G. 
34 


898  NOTES. 

which  a  given  number  bears  to  unity  can  not,  indeed,  be  the 
subject  of  inquiry ;  because  it  is  expressed  by  the  number 
itself.  But  the  ratio  which  it  bears  to  othir  numbers  may  be 
as  proper  an  object  of  mathematical  investigation,  as  the 
ratio  of  a  mile  to  a  furlong. 

For  proof  that  number  is  not  quantity,  Mr.  Stewart  refers 
to  Barrow's  Mathematical  Lectures.  Dr.  Barrow  has  started 
an  etymological  objection  to  the  application  of  the  term 
quantity  to  number,  which  he  intimates  might,  with  more 
propriety,  be  called  quotity.  He  observes,  "  The  general 
object  of  the  mathematics  has  no  proper  name,  either  in  Greek 
or  Latin."  And  adds,  "  It  is  plain  the  mathematics  is  con- 
versant about  two  things  especially,  quantity  strictly  taken, 
and  quotity ;  or  magnitude  and  multitude."  There  is  fre- 
quent occasion  for  a  common  name,  to  express  number,  dura- 
tion, &c.  as  well  as  magnitude ;  and  the  term  quantity  will 
probably  be  used  for  this  purpose,  till  some  other  word  is  sub- 
stituted in  its  stead. 

But  though  Dr.  Barrow  thus  distinguishes  between  mag- 
nitude and  number,  he  afterwards  gives  it  as  his  opinion, 
J  page  20,  49,)  that  there  is  really  no  quantity  in  nature  (lif- 
erent from  what  is  called  magnitude  or  continued  quantity, 
and  consequently  that  this  alone  ought  to  be  accounted  the 
object  of  the  mathematics.  He  accordingly  devotes  a  whole 
lecture  to  the  purpose  of  proving  the  identity  of  arithmetic 
and  geometry.  (Lect.  3.)  He  is  "convinced  that  number 
really  differs  nothing  from  what  is  called  continued  quantity  ; 
but  is  only  formed  to  express  and  declare  it ;"  that  as  "  the 
conceptions  of  magnitude  and  number  could  scarcely  be  sepa- 
rated," by  the  ancients,  "  in  the  name,  they  can  hardly  be  so 
in  the  mind"  and  "  that  number  includes  in  it  every  conside- 
ration pertaining  to  geometry/'  He  admits  of  metaphysical 
number,  which  is  not  the  object  of  geometry,  or  even  of  the 
mathematics.  But,  in  his  view,  magnitude  is  always  inclu- 
ded in  mathematical  number,  as  the  units  of  which  it  is  com- 
posed are  equal.  On  the  other  hand,  magnitudes  are  not  to 
be  considered  as  mathematical  quantities,  except  as  they  are 
measured  by  number.  In  short,  quantity  is  magnitude  meas- 
ured by  number. 

It  would  seem,  then,  that  according  to  Dr.  Barrow,  num- 
ber considered  as  separate  from  magnitude,  has  as  fair  a 
claim  to  be  cal'ed  quantity,  as  magnitude  considered  as  sepa 
rate  from  number.    If  arithmetic  and  geometry  are  the  same ; 


NOTES.  39D 

quantity  is  as  much  the  object  of  one,  as  of  the  other.  How 
far  this  scheme  is  applicable  to  duration,  motion,  &c.  it  is  not 
necessary,  in  this  place,  to  inquire. 

Note  B.  p.  35. 

It  is  common  to  define  multiplication,  by  saying  that  '  it  is 
finding  a  product  which  has  the  same  ratio  to  the  multipli- 
cand, that  the  multiplier  has  to  a  unit/  This  is  strictly  and 
universally  true.  But  the  objection  to  it,  as  a  definition,  is, 
that  the  idea  of  ratio,  as  the  term  is  understood  in  arithmetic 
and  algebra,  seems  to  imply  a  previous  knowledge  of  multi- 
plication, as  well  as  of  division.  In  this  work  at  least,  the 
expression  of  geometrical  ratio  is  made  to  depend  on  division, 
and  division  on  multiplication.  Ratio,  therefore,  could  not 
be  properly  introduced  into  the  definition  of  multiplication. 

It  is  thought,  by  some,  to  be  absurd  to  speak  of  a  unit  as 
consisting  of  parts.  But  whatever  may  be  true  with  respect 
to  number  in  the  abstract,  there  is  certainly  no  absurdity  in 
considering  an  integer,  of  one  denomination,  as  made  up  of 
parts  of  a  different  denomination.  One  rod  may  contain 
several  feet :  one  foot  several  inches,  &c.  And  in  multipti- 
cation,  we  may  be  required  to  repeat  the  whole,  or  a  part  oT 
the  multiplicand,  as  many  times  as  there  are  inches  in  a  foot, 
or  part  of  a  foot. 

Note  C.   p.  55. 

Strictly  speaking,  the  inquiry  to  be  made  is,  how  often  ths 
whole  divisor  is  contained  in  as  many  terms  of  the  dividend. 
But  it  is  easier  to  divide  by  a  part  only  of  the  divisor;  and 
this  will  lead  to  no  error  in  the  result,  as  the  whole  divisor  is 
multiplied,  in  obtaining  the  several  subtrahends. 

Note  D.  p.  86. 

It  is  perhaps  more  philosophically  exact,  to  consider  an 
equation  as  affirming  the  equivalence  of  two  different  expres- 
sions of  the  same  quantity,  than  to  speak  of  it  as  expressing 
an  equality  between  one  quantity  and  another.  But  it  is 
doubted  whether  the  former  definition  is  the  best  adapted  to 
the  apprehension  of  the  learner;  who  in  this  early  part  of  his 
mathematical  course,  may  be  supposed  to  be  very  little  ac- 
customed to  abstraction.  Though  he  may  see  clearly,  that 
the  area  of  a  triangle  is  equal  to  the  area  of  a  parallelogram 


400  NOTES.    . 

of  the  same  base  and  half  the  height ;  yet  he  may  hesitate  in 
pronouncing  that  the  two  surfaces  are  precisely  the  same. 

Note  E.  p.  129. 

As  the  direct  powers  of  an  integral  quantity  have  positive 
indices,  while  the  reciprocal  powers  have  negative  indices  ;  it 
is  common  to  call  the  former  positive  powers,  and  the  latter 
negative  powers.  But  this  language  is  ambiguous,  and  may 
lead  to  mistake.  For  the  same  terms  are  applied  to  powers 
with  positive  and  negative  signs  prefixed.  Thus  +8a4  is 
called  a  positive  power;  while  —8a*  is  called  a  negative 
one.  It  may  occasion  perplexity,  to  speak  of  the  latter  as 
being  both  positive  and  negative  at  the  same  time ;  positive, 
because  it  has  a  positive  index,  and  negative  because  it  has 
a  negative  co-efficient.  This  ambiguity  may  be  avoided, 
by  using  the  terms  direct  and  reciprocal ;  meaning,  by  the 
former,  powers  with  positive  exponents,  and  by  the  latter, 
powers  with  negative  exponents. 

Note  F.  p.  202. 

For  the  sake  of  keeping  clear  of  the  multiplied  controver- 
sies, a  great  portion  of  them  verbal,  respecting  the  nature  of 
ratio,  I  have  chosen  to  define  geometrical  ratio  to  be  that 
which  is  expressed  by  the  quotient  of  one  quantity  divided  by 
another,  rather  than  to  say  that  it  consists  in  this  quotient. 
Every  ratio  which  can  be  mathematically  assigned,  may  be 
expressed  in  this  way,  if  we  include  surd  quantities  among 
those  which  are  to  be  admitted  into  the  numerator  or  de- 
nominator of  the  fraction  representing  the  quotient. 

Note  G.  p.  204. 

This  definition  of  compound  ratio  is  more  comprehensive 
than  the  one  which  is  given  in  Euclid.  That  is  included  in 
this,  but  is  limited  to  a  particular  case,  which  is  stated  in 
Art.  357.  It  may  answer  the  purposes  of  geometry,  but  is 
not  sufficiently  general  for  algebra. 

Note  H.  p.  206. 

It  is  not  denied  that  very  respectable  writers  use  these 
terms  indiscriminately.  But  it  appears  to  be  without  any 
necessity.     The  ratio  of  6  to  2  is  3.     There  is  certainly  a 


VOTES  401 

difference  between  twice  this  ratio,  and  the  square  of  it,  that 
is,  between  twice  three,  and  the  square  of  three.  All  are 
agreed  to  call  the  latter  a  duplicate  ratio.  What  occasion  is 
there,  then,  to  apply  to  it  the  term  double  also?  This  is 
wanted,  to  distinguish  the  other  ratio.  And  if  it  is  confined 
to  that,  it  is  used  according  to  the  common  acceptation  of 
the  word,  in  familiar  /anguage. 

Note  I.  p.  214. 

The  definition  here  given  is  meant  to  be  applicable  to 
quantities  of  every  description.  The  subject  of  proportion 
as  it  is  treated  of  in  Euclid,  is  embarrassed  by  the  means 
which  are  taken  to  provide  for  the  case  of  incommensurable 
quantities.  But  this  difficulty  is  avoided  by  the  algebraic 
notation  which  may  represent  the  ratio  even  of  incommen- 
surables. 

Thus  the  ratio  of  1  to   </2  is  — ~ 

It  is  impossible,  indeed,  to  express  in  rational  numbers,  the 
square  root  of  2,  or  the  ratio  which  it  bears  to  1.  But  this 
is  not  necessary,  for  the  purpose  of  showing  its  equality  with 
another  ratio. 

The  product  4X2=8. 

And,  as  equal  quantities  have  equal  roots, 

2X  v/2=  v/8,  therefore,  2  :  </8  : :  1 :  >/2. 

Here  the  ratio  of  2  to  v/8,  is  proved  to  be  the  same,  as 
that  of  1  to  y/2 ;  although  we  are  unable  to  find  the  exact 
value  either  of  v/8  or  v/2. 

It  is  impossible  to  determine,  with  perfect  accuracy,  the 
ratio  which  the  side  of  a  square  has  to  its  diagonal.  Yet  it 
is  easy  to  prove,  that  the  side  of  one  square  has  the  same 
ratio  to  its  diagonal,  which  the  side  of  any  other  square  has 
to  its  diagonal.  When  incommensurable  quantities  are  once 
reduced  to  a  proportion,  they  are  subject  to  the  same  laws  as 
other  proportionals.  Throughout  the  section  on  proportion, 
the  demonstrations  do  not  imply  that  we  know  the  value  of 
the  terms,  or  their  ratios ;  but  only  that  one  of  the  ratios  is 
equal  to  the  other. 

34* 


402  NOTES. 

Note  K.  p.  218. 

The  inversion  of  the  means  can  be  made  with  strict  pro- 
priety in  those  cases  only  in  which  all  the  terms  are  quanti- 
ties of  the  same  kind.  For,  if  the  two  last  be  different  from 
the  two  first,  the  antecedent  of  each  couplet,  after  the  inver- 
sion will  be  different  from  the  consequent,  and  therefore, 
there  can  be  no  ratio  between  them.     (Art.  359.) 

This  distinction,  however,  is  of  little  importance  in  prac- 
tice. For,  when  the  several  quantities  are  expressed  in 
numbers,  there  will  always  be  a  ratio  between  the  numbers. 
And  when  two  of  them  are  to  be  multiplied  together,  it  is 
immaterial  which  is  the  multiplier,  and  which  is  the  multipli- 
cand. Thus  in  the  Rule  of  Three  in  arithmetic,  a  change 
in  the  order  of  the  two  middle  terms  will  make  no  difference 
in  the  result. 

Note  L.   p.  225. 

The  terms  composition  and  division  are  derived  from  ge- 
ometry, and  are  introduced  here,  because  they  are  generally 
used  by  writers  on  proportion.  But  they  are  calculated  rather 
to  perplex,  than  to  assist  the  learner.  The  objection  to  the 
word  composition  is,  that  its  meaning  is  liable  to  be  mistaken 
for  the  composition  or  compounding  of  ratios.  (Art.  398.) 
The  two  cases  are  entirely  different,  and  ought  to  be  care- 
fully distinguished.  In  one,  the  terms  are  added,  in  the  other, 
they  are  multiplied  together.  The  word  compound  has  a 
similar  ambiguity  in  other  parts  of  the  mathematics.  The 
expression  a -\-b,  in  which  a  is  added  to  b,  is  called  a  com- 
pound quantity.  The  fraction  \  of  f,  or  ^X§,  in  which  ^  is 
multiplied  into  |,  is  called  a  compound  fraction. 

The  term  division,  as  it  is  used  here,  is  also  exceptionable. 
The  alteration  to  which  it  is  applied,  is  effected  by  subtrac- 
tion, and  has  nothing  of  the  nature  of  what  is  called  division 
in  arithmetic  and  algebra.  But  there  is  another  case,  (Art. 
400,)  totally  distinct  from  this,  in  which  the  change  in  the 
terms  of  the  proportion  is  actually  produced  by  division. 

Note  M.    p.  234. 

The  principles  stated  in  this  section,  are  not  only  expressed 
in  different  language,  from  the  corresponding  propositions  in 
Euclid,  but  are  in  several  instances  more  general.  Thus  the 
first  proposition,  in  the  fifth  book  of  the  Elements,  is  confined 


NOTES.  403 

to  equimultiples.  But  the  article  referred  to,  as  containing  this 
proposition,  is  applicable  to  all  cases  of  equal  ratios,  whether 
the  antecedents  are  multiples  of  the  consequents  or  not. 

Note  N.  p.  250. 

The  solution  of  one  of  the  cases  is  omitted  in  the  text,  be- 
cause it  is  performed  by  logarithms,  with  which  the  learner 
is  supposed  not  to  be  acquainted,  in  this  part  of  the  course. 
When  the  first  term,  the  last  term,  and  the  ratio  are  given, 
the  number  of  terms  may  be  found  by  the  formula 

rz 

log.  — 

b    a 

log.  r 

Note  O.  p.  255. 

When  it  is  said  that  a  mathematical  quantity  may  be  sup- 
posed to  be  increased  beyond  any  determinate  limits,  it  is  not 
intended  that  a  quantity  can  be  specified  so  great,  that  no 
limits  greater  than  this  can  be  assigned.  The  quantity  and 
the  limits  may  be  alternately  extended  one  beyond  the  other. 
If  a  line  be  conceived  to  reach  the  most  distant  point  in  the 
visible  heavens,  a  limit  may  be  mentioned  beyond  this.  The 
line  may  then  be  supposed  to  be  extended  farther  than  this 
limit.  Another  point  may  be  specified  still  farther  on,  and 
yet  the  line  may  be  conceived  to  be  carried  beyond  it. 

Note  P.  p.  257. 

The  apparent  contradictions  respecting  infinity,  are  owing 
to  the  ambiguity  of  the  term.  It  is  often  thought  that  the 
proposition,  that  quantity  is  infinitely  divisible,  involves  an 
absurdity.  If  it  can  be  proved  that  a  line  an  inch  long  can 
be  divided  into  an  infinite  number  of  parts,  it  can,  by  the 
same  mode  of  reasoning,  be  proved,  that  a  line  two  inches 
long  may  be  first  divided  in  the  middle,  and  then  each  of  the 
sections  be  divided  into  an  infinite  number  of  parts.  In  this 
way,  we  shall  obtain  one  infinite  twice  as  great  as  another. 

If  by  infinity,  here  is  meant  that  which  is  beyond  any 
assignable  limits,  one  of  these  infinites  may  be  supposed 
greater  than  the  other,  without  any  absurdity.  But  if  it  be 
meant  that  the  number  of  divisions  is  so  great  that  it  can  not 
be  increased,  we  do  not  prove  this,  concerning  either  of  the 


404  NOTES. 

lines.  We  make  out,  therefore  no  contradiction.  The  ap- 
parent absurdity  arises  trom  shifting  the  meaning  of  the  terms. 
We  demonstrate  that  a  quantity  is,  in  one  sense  infinite;  and 
then  infer  that  it  is  infinite,  in  a  sense  widely  different. 

Note  Q.  p.  386. 

It  will  be  thought,  perhaps,  that  it  was  unnecessary  to  be 
so  particular,  in  obtaining  the  expression  for  the  area  of  a 
parallelogram,  for  the  use  of  those  who  read  Playfair's  edi- 
tion of  Euclid,  in  which  "  AD. DC  is  put  for  the  rectangle 
contained  by  AD  and  DC"  Jt  is  to  be  observed,  however, 
that  he  introduces  this,  merely  as  an  article  of  notation, 
(Book  II,  Def.  1.)  And  though  a  point  interposed  between 
the  letters,  is,  in  algebra,  a  sign  of  multiplication  ;  yet  he  does 
not  here  undertake  to  show  how  the  sides  of  a  parallelogram 
may  be  multiplied  together.  In  the  first  book  of  the  Supple- 
ment, he  has  indeed  demonstrated,  that  "equiangular  paral- 
lelograms are  to  one  another,  as  the  products  of  the  numbers 
proportional  to  their  sides."  But  he  has  not  given  to  the  ex- 
pressions the  forms  most  convenient  for  the  succeeding  parts 
of  this  work.  In  making  the  transition  from  pure  geometry 
to  algebraic  solutions  and  demonstrations,  it  is  important  to 
have  it  clearly  seen  that  the  geometrical  principles  are  not 
altered ;  but  are  only  expressed  in  a  different  language. 

Note  R.  p.  3Q5. 

This  section  comprises  very  little  of  what  is  commonly 
understood  by  the  application  of  algebra  to  geometry.  The 
principal  object  has  been,  to  prepare  the  way  for  the  other 
parts  of  the  course,  by  stating  the  grounds  of  the  algebraic 
notation  of  geometrical  quantities,  and  rendering  it  familiar 
by  a  few  examples. 


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